Homotopy classification of maps between A 2 n -complexes
aa r X i v : . [ m a t h . A T ] A ug HOMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES PENGCHENG LI
Abstract.
This article is partial work of the author’s PhD thesis. In this article, the author improvesBauess homotopy classification of maps between indecomposable ( n − ) -connected ( n + ) dimensionalfinite CW-complexes X, Y , n > 3 , by finding a generating set for any abelian group [ X, Y ] .Using these generators, the author firstly finds splitting cofiber sequences which imply Zhu-Pansdecomposability result of smash products of the complexes, and secondly, obtains partial results onthe groups of homotopy classes of self-homotopy equivalences of the complexes and some of theirnatural subgroups. Contents
1. Introduction 12. Preliminaries 72.1. Baues’s diagrams and cofiber sequences for Chang-complexes 72.2. Spanier-Whitehead duality 82.3. The Steenrod squares 83. Homotopy Classification of Maps between A -complexes 83.1. Proof of Table 1 93.2. Proof of Table 2 144. Applications 224.1. The decomposability of smash products of A -complexes 224.2. The groups of self-homotopy equivalences of A -complexes 28Acknowledgements 37References 371. Introduction
In this article we work in the category
FCW ∗ of based finite CW-complexes with morphism set FCW ∗ ( X, Y ) = [
X, Y ] , the the set of the homotopy classes of base-point-preserving maps from X to Y ;we don’t distinguish a map f : X → Y with the homotpy class [ f ] ∈ [ X, Y ] and denote by X ≃ Y if X ishomotopic to Y in FCW ∗ . Let A kn be the set consisting of ( n − ) -connected spaces with dimensionat most n + k . By the generalized Freudenthal suspension theorem(Theorem 1.21 of [11]), for A kn -complexes X, Y , the suspension map Σ : [ X, Y ] → [ ΣX, ΣY ] is a bijection if n > k + and is a surjectionif n ≥ k + ; consequently, X, Y are stable if n ≥ k + and a map f : X → Y is stable if n > k + .The classification of homotopy types of indecomposable A kn -complexes is a classical problem inhistory. Recall that a space X is decomposable if there exist non-contractible spaces X and X such that X ≃ X ∨ X ; otherwise X is called indecomposable . As a direct application of classical homotopy theory,it is a basic fact that indecomposable A ( n ≥ ) -complexes are just spheres S n and indecomposable A ( n ≥ ) -complexes consist of spheres S n , S n + and Moore spaces M np r = S n ∪ p r C S n with p a Mathematics Subject Classification.
Primary 55P15, 55P42, 55P05, 55P25, 55P10.
Key words and phrases. homotopy classification, A -complexes, stable homotopy, self-homotopy equivalences. Email address : [email protected] . prime and r a positive integer. In 1950, S. Chang [9] firstly showed that indecomposable A ( n ≥ ) -complexes can be classified into the following three types: ( a ) Spheres: S n + i , i =
0, 1, 2 ; ( b ) Elementary Moore spaces: M n + jp r = M ( Z /p r , n + j ) = S n + j ∪ p r C S n + j , where p is a prime, j =
0, 1 and r ∈ Z + , the set of positive integers; ( c ) Elementary Chang-complexes: C n + , C n + , C n + , C n + , where r, t ∈ Z + .The cell structures of elementary Chang-complexes can be described as follows: • C n + = S n S η C S n + , • C n + = ( S n ∨ S n + ) S(cid:16) η2 t (cid:17) C S n + = S n S ηq C M n2 t ; • C n + = S n S ( r ,η ) C ( S n ∨ S n + ) = M n2 r S iη C S n + ; • C n + = ( S n ∨ S n + ) S(cid:16) r , η0 , 2 t (cid:17) C ( S n ∨ S n + ) = ( M n2 r ∨ S n ) S(cid:16) iη2 t (cid:17) C S n + = S n S ( r ,ηq ) C ( S n ∨ M n2 t ) = M n2 r S iηq C M n2 t ;where i = i n : S n → M n2 r , q = q n + : M n2 t → S n + are the canonical inclusion and projection maps,respectively; η = η n : S n + → S n is the iterated suspensions of the Hopf map.In 1991 Baues and Hennes [7] classified the indecomposable homotopy types of A ( n ≥ ) -complexes; later in 2004 Drozd [12] proved that the classification problem of the homotopy types of A kn -complexes for k ≥ is “wild” in the sense similar to that in representation of finite dimensionalalgebras.Another classical problem is to classify the homotopy classes of maps between A kn -complexes inthe stable range. The case k = is clear and the case k = was firstly computed by Brown andCopeland [8]. In 1985 Baues [6] listed the group structures [ X, Y ] of elementary A -complexes withouta proof, possibly due to the layout of the book. In this article, the author gives a new proof of thegroup structures [ X, Y ] and improves Baues’s results by choosing explicit generators of these groups,see Section 3. Remark . The importance of generators of maps between A -complexes includes but is not limitedto: Determine the group structures and the corresponding generators of the sets of maps between the(reduced) smash products of A -complexes. From Theorem 4.5, we see that there are splitting cofibrations for some smash products of elementary Chang-complexes, which improves Zhu andPan results about the decomposability of smash products of A -complexes, see [25]. Obtain an explicit expression of the switching map T : X ∧ X → X ∧ X of the suspension spectrum X of an elementary Chang-complex X , as Toda did in his papers [1, 2] for the Moore spectrum M q with ( M q ) n = M ( Z /q, n ) . Furthermore, generators of maps between elementary A -complexesplay a key role in studying the possible ring spectrum structure on the suspension spectrum C rr ,since Theorem 4.5 implies that there is a splitting cofibration of suspension spectra: S ∧ C rr i ∧ C . . C rr ∧ C rr q C ∧ C / / τ n n C r ∧ C rr , where S n + = S n + , ( C rr ) n + = C n + , ( C r ) n + = C n + for each n ≥ . The work of the ringspectrum structure on the Moore spectrum M q is mainly due to Oda, see [16, 17, 18]. Compute the groups E ( X ) of (homotopy classes of) self-homotopy equivalences of Chang-complexesand their subgroups, see Section 4.2.We now give notations needed to make sense of the introduction and characterize generators ofmaps between elementary A -complexes. Let X be an A -complex, n ≥ .
1. 1 X denotes the identity of X ; particularly, n = S n , M = M n2r , η = C n + . If there is no confu-sion, we denote by C = C n + , C = C n + , C = C n + for different subscripts and superscripts. OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 3
2. i and q denote the canonical inclusion and projection maps, respectively. If there is no confusionand for simplicity, we use the same notations for different inclusions and projections: i ) i n + k and q n + k denote the possible canonical inclusions: S n + k → X and the possible canonicalprojections: X → S n + k , respectively. For example, we denote by i n : S n → X the canonicalinclusions for different spaces X = M n2 r , C n + , C n + , C n + , C n + . ii ) i M and q M denote the possible inclusions M n2 r → X and projections X → M n + t , respectively. iii ) i η : C n + → C n + , i C : C n + → C n + denote the canonical inclusions and q η : C n + → C n + , q C : C n + → C n + denote the canonical projections by collapsing the subspaces S n + .Note that there hold equalities of compositions of maps: i M i n = i n , i η i n = i n , i C i n = i n ; q n + q M = q n + , q n + q C = q n + , q n + q η = q n + .3. η ∈ π s1 , ν ∈ π s3 , σ ∈ π s7 denote iterated suspensions of the Hopf maps. Note η = · ν . Let A = A h a , a , · · · i = A { a , a , · · · } be an abelian group with a generator set { a , a , · · · } . H n ( X ; A ) denotes the reduced homology group of X with coefficient group A ; if A = Z , write H n ( X ; Z ) = H n ( X ) . Similar notations are adopted in the cohomology case. For a map f : X → Y , we denote the induced homomorphisms on homology groups and cohomology groups by f ∗ = H n ( f ; A ) : H n ( X ; A ) → H n ( Y ; A ) and f ∗ = H n ( f ; A ) : H n ( Y ; A ) → H n ( X ; A ) , respectively.
5. B n ( χ ) : M np r → M np r ′ is characterized by the following two properties [5, p.199]: ( i ) H n ( B n ( χ )) = χ : Z /p r → Z /p r ′ satisfies: χ ( ) = (cid:12)
1, r ≥ r ′ ; p r ′ − r , r < r ′ . ( ii ) B n ( χ ) = Σ n − B ( χ ) , where Σ n − : [ M r , M r ′ ] → [ M np r , M np r ′ ] is the composition of ( n − ) suspension maps.Here we adopt the same notations as Baues for different r, r ′ if there is no confusion. An alternativenotation is B nr ′ ,r = B n ( χ ) , which satisfies relations in Proposition 1.2 ( ) .The article is arranged as follows. Section 2 will review some preliminary knowledge in algebraictopology, including the cofiber sequences of elementary Chang-complexes, Spanier-Whitehead duality.In Section 3 we shall give a new proof of the homotopy classification of maps between A ( n ≥ ) -complexes and choose their appropriate generators, which are summarized in Table 1 and Table 2.By [6], if p is an odd prime, the non-trivial groups [ M n + ip r , X ] with X an elementary Chang-complexhappens only if i = and X = C n + , C n + . As can be seen from these two tables, only the case p = are covered and most generators are compositions of the canonical maps defined above; furthermore,the new ones are extensions or lifts of the known generators. Proposition 1.2.
Let n ≥ . ˜ η r : S n + → M n2 r and ¯ η r ∈ [ M n + r , S n ] satisfy the relations: q n + ˜ η r = η = ¯ η r i n + ; ¯ η = η q n + , 2 ˜ η = i n η . (1.1)
2. B n ( χ ) = B nr ′ ,r ∈ [ M n2 r , M n2 r ′ ] satisfies B nr,r = M , ΣB n ( χ ) = B n + ( χ ) and the relations: r ≥ r ′ : B n ( χ ) i n = i n , q n + B n ( χ ) = r − r ′ · q n + ; r < r ′ : B n ( χ ) i n = r ′ − r · i n , q n + B n ( χ ) = q n + . (1.2) ˜ ζ ∈ [ S n + , C n + ] and ¯ ζ ∈ [ C n + , S n ] satisfy the relations: q n + ˜ ζ = · n + , ¯ ζi n = · n , ˜ ζq n + + i n ¯ ζ = · η ; ¯ ζ ˜ ζ = η . (1.3) PENGCHENG LI ˜ ξ ∈ [ M n2 t + , C n + ] and ¯ ξ ∈ [ C n + , M n2 r + ] satisfy the relations: ˜ ξi n + = i n + , q η ˜ ξ = ˜ ζq n + , q M ˜ ξ = B n + ( χ ) , q n + ˜ ξ = n + ; q n + ¯ ξ = q n + , ¯ ξi η = i n ¯ ζ, ¯ ξi M = B n ( χ ) , ¯ ξi n = n . (1.4) If t ≤ t ′ , ˜ κ ∈ [ C n + , C n + ′ ] satisfy the relations: ˜ κ · i n = i n , q ′ M ˜ κ = B n + ( χ ) q M , q η ˜ κ = q η ; ˜ κ = C ( t = t ′ ) . (1.5) If r ≥ r ′ , ¯ κ ∈ [ C n + , C n + ′ ] satisfy the relations: q n + ¯ κ = q n + , ¯ κi M = i ′ M B n ( χ ) , ¯ κi η = i η ; ¯ κ = C ( r = r ′ ) . (1.6)
6. q C i C = i η q η ; if t ′ ≥ t, r ≥ r ′ , B n ( χ ) = i C ˜ κ ∈ [ C n + , C n + ′ r ′ ] satisfy the relations: B n ( χ ) i C = i C ˜ κ, B n ( χ ) i M = i ′ M B n ( χ ) , q ′ M B n ( χ ) = B n + ( χ ) q M . (1.7) Proof. refers to [7], by the definition of B n ( χ ) ; follows from the relations ( ) of [2]. Theproof of ∼ will be completed in Section 3. (cid:3) In Section 4 we firstly apply the generators of maps between A -complexes to improve Zhu andPan’s decomposability results of smash products of Chang-complexes by finding the correspondingsplitting cofiber sequences in the decomposable cases. A known cheif application of the splittingcofiber sequence for C n + ∧ C n + is the following published result: Theorem 1.3 (Theorem 1.1 of [24]) . For n ≥ , there is a homotopy decomposition: ΩΣC n + ≃ Y j ΩΣC k j ( n + )+ × (some other space) , where < k < · · · is a sequence of odd integers such that k j is not a multiple of any k i else foreach j . The remainder of Section 4 covers partial results on the groups of self-homotopy equivalences of A -complexes and some of their natural subgroups. Especially, the author discussed the relationsbetween the subgroup E ∗ ( C n + r ∨ C n + r ) of self-homotopy equivalences that induces the identityin homology and the subgroups E n + l ♯ ( C n + r ∨ C n + r )( l =
1, 2 ) of self-homotopy equivalences thatinduces the identity in homotopy groups of dimensions ≤ n + l in Section 4.2.1. O M O T O P Y C L A SS I F I C A T I O N O F M A P S B E T W EE N A n - C O M P L E X E S Table 1.
Maps between Moore spaces and Chang-complexes S n S n + S n + M n2 r M n + t S n Z n Z /2 η Z /2 η Z /2 ηq n + t = : Z /4 ¯ η t > 1 : Z /2 ⊕ Z /2 ¯ η t , η q n + S n + Z n + Z /2 η Z /2 r q n + Z /2 ηq n + S n + Z n + Z /2 t q n + M n2 r ′ Z /2 r ′ i n Z /2 i n η r ′ = : Z /4 ˜ η r ′ > 1 : Z /2 ⊕ Z /2 ˜ η r ′ , i n η r = r ′ = : Z /4 1 M otherwise: Z /2 l ⊕ Z /2B n ( χ ) , i n ηq n + t = = r ′ : Z /2 ⊕ Z /2i n ¯ η , ˜ η q n + t > 1 = r ′ : Z /2 ⊕ Z /4i n ¯ η t , ˜ η q n + t = ′ : Z /4 ⊕ Z /2i n ¯ η , ˜ η r ′ q n + t > 1 < r ′ : Z /2 ⊕ Z /2 ⊕ Z /2i n ¯ η t , ˜ η r ′ q n + , i n η q n + M n + t ′ Z /2 t ′ i n + Z /2 i n + η Z /2 m i n + q n + t = t ′ = : Z /4 1 M otherwise: Z /2 n ⊕ Z /2B n + ( χ ) i n + ηq n + C n + Z i n Z ˜ ζ 0 Z /2 t ˜ ζq n + C n + ′ Z i n Z /2 t ′ + i n + Z /2 i n + η Z /2 m ′′ i n + q n + Z /2 n ′′ ⊕ Z /2 ˜ ξB n + ( χ ) , i n + ηq n + C n + ′ Z /2 r ′ i n Z ⊕ Z /2i η ˜ ζ, i M ˜ η r ′ Z /2 l i M B n ( χ ) Z /2 t ⊕ Z /2i η ˜ ζq n + , i M ˜ η r ′ q n + C n + ′ r ′ Z /2 r ′ i n Z /2 t ′ + i n + Z /2 ⊕ Z /2i n + η, i M ˜ η r ′ Z /2 l ⊕ Z /2 m ′′ i M B n ( χ ) , i n + q n + Z /2 n ′′ ⊕ Z /2 ⊕ Z /2i C ˜ ξB n + ( χ ) , i M ˜ η r ′ q n + , i n + ηq n + j = max ( t, r ′ ) , k = min ( t, r ′ ); l = min ( r, r ′ ) , l ′ = min ( r +
1, r ′ ); m = min ( r, t ′ ) , m ′ = min ( r +
1, t ′ ) , m ′′ = min ( r, t ′ + ); n = min ( t, t ′ ) , n ′′ = min ( t, t ′ + ) PE N G C H E N G L I Table 2.
Maps between Chang-complexes C n + C n + C n + C n + S n Z ¯ ζ Z ¯ ζq η ⊕ Z /2 ¯ η t q M Z /2 ηq n + Z /2 ⊕ Z /2 ¯ η t q M , ηq n + S n + Z /2 r + q n + Z /2 r + q n + S n + Z q n + Z /2 t q n + Z q n + Z /2 t q n + M n2 r ′ Z /2 r ′ i n ¯ ζ Z /2 r ′ ⊕ Z /2i n ¯ ζq η , i n ¯ η t q M Z /2 l ′ ⊕ Z /2B n ( χ ) ¯ ξ, i n ηq n + Z /2 l ′ ⊕ Z /2 ⊕ Z /2B n ( χ ) ¯ ξq C , i n ¯ η t q M , i n ηq n + M n + t ′ Z /2 n B n + ( χ ) q M Z /2 m ′ i n + q n + Z /2 m ′ ⊕ Z /2 n i n + q n + , B n + ( χ ) q M C n + Z ⊕ Z η , i n ¯ ζ Z ⊕ Z /2 t q η ˜ ζq n + Z ˜ ζq n + Z /2 t ˜ ζq n + C n + ′ Z i n ¯ ζ t > t ′ : Z ⊕ Z /2 t ′ + i n ¯ ζq η , ˜ ξB n + q M t ≤ t ′ : Z ⊕ Z /2 t ˜ κ, ˜ ξB n + ( χ ) q M Z /2 m + i n + q n + Z /2 n ′′ ⊕ Z /2 m + ˜ ξB n + ( χ ) q M , i n + q n + C n + ′ Z ⊕ Z /2 r ′ i η , i n ¯ ζ t ≥ r ′ : Z /2 t + ⊕ Z /2 r ′ i η q η , i n ¯ ζq η t < r ′ : Z /2 r ′ + ⊕ Z /2 t i η q η , i η ˜ ζq n + r ′ > r : Z ⊕ Z /2 r + i η ˜ ζq n + , i M B n ( χ ) ¯ ξr ′ ≤ r : Z ⊕ Z /2 r ′ ¯ κ, i M B n ( χ ) ¯ ξ r ≥ r ′ ≤ t : Z /2 t + ⊕ Z /2 r ′ ¯ κq C , i M B n ( χ ) ¯ ξq C r ≥ r ′ > t : Z /2 r ′ + ⊕ Z /2 t ¯ κq C , i η ˜ ζq n + r < r ′ : Z /2 r + ⊕ Z /2 t i M B n ( χ ) ¯ ξq C , i η ˜ ζq n + C n + ′ r ′ Z /2 r ′ i n ¯ ζ t ′ ≥ t < r ′ : Z /2 r ′ + ⊕ Z /2 t i C ˜ κ, i C ˜ ξB n + ( χ ) q M t ′ ≥ t ≥ r ′ : Z /2 t + ⊕ Z /2 r ′ i C ˜ κ, i n ¯ ζq η t ′ < t : Z /2 t ′ + ⊕ Z /2 r ′ i C ˜ ξB n + ( χ ) q M , i n ¯ ζq η Z /2 m + ⊕ Z /2 l ′ i n + q n + , i M B n ( χ ) ¯ ξ r ′ > r ∨ t ′ < t : Z /2 m + ⊕ Z /2 l ′ ⊕ Z /2 n ′′ i n + q n + , i M B n ( χ ) ¯ ξq C , i C ˜ ξB n + ( χ ) q M t ′ ≥ t < r ′ ≤ r : Z /2 m + ⊕ Z /2 r ′ + ⊕ Z /2 t i n + q n + , i C ˜ κ, i C ˜ ξB n + ( χ ) q M t ′ ≥ t ≥ r ′ ≤ r : Z /2 m + ⊕ Z /2 t + ⊕ Z /2 r ′ i n + q n + , i C ˜ κ, i M B n ( χ ) ¯ ξq C j = max ( t, r ′ ) , k = min ( t, r ′ ); l = min ( r, r ′ ) , l ′ = min ( r +
1, r ′ ); m = min ( r, t ′ ) , m ′ = min ( r +
1, t ′ ) , m ′′ = min ( r, t ′ + ); n = min ( t, t ′ ) , n ′′ = min ( t, t ′ + ) OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 7 Preliminaries
Baues’s diagrams and cofiber sequences for Chang-complexes.
The cell structures ofelementary Chang-complexes can be intuitively described by Baues’s gluing diagrams: ✁✁✁✁✁✁ • n • n + n + η ✁✁✁✁✁✁ • n • n + n + • r ✁✁✁✁✁✁ • n • η n + n + • t ✁✁✁✁✁✁ • n η • n + n + • r • t In the above diagrams, the bullets with dimensions labelled nearby mean the cells of the labelleddimensions and the lines indicate the attaching maps; in particular, the slant lines all denote thesuspended Hopf map η = η n : S n + → S n . For brevity we omit the labels of the middle ’bullets’,whose dimensions are n + . Note that the bullets of dimension n denote S n , not n -cells.The following cofiber sequences for elementary Chang complex is due to Zhu and Pan [25].Let i S = ( i n , i n + ) , i M = ( i M , i n ) ; q S = (cid:18) q n + q n + (cid:19) , q M = (cid:18) q n + q M (cid:19) . • The cofibre sequence for C n + Cof1 : S n + −− → S n i n −− → C n +
2η q n + −−− → S n + → S n + • The cofibre sequences for C n + Cof1 : S n ∨ S n + ( r ,η ) −−−− → S n i n −− → C n +
2r q S −− → S n + ∨ S n + → S n + ; Cof2 : S n + n η −−− → M n2 r i M −− → C n +
2r q n + −−−− → S n + → M n + r ; Cof3 : S n i n r −−− → C n +
2η i η −− → C n +
2r q n + −−−− → S n + → C n + ; • The cofibre sequences for C n + Cof1 : S n + (cid:16) η2 t (cid:17) −−−−− → S n ∨ S n + S −− → C n + n + −−−− → S n + → S n + ∨ S n + ; Cof2 : M n2 t ηq n + −−−−− → S n i n −− → C n + M −− → M n + t → S n + ; Cof3 : C n +
1η 2 t q n + −−−−− → S n + n + −−− → C n + η −− → C n + → S n + ; • The cofibre sequences for C n + Cof1 : S n ∨ S n + (cid:16) r , η0 , 2 t (cid:17) −−−−−−− → S n ∨ S n + S −− → C n + S −− → S n + ∨ S n + → S n + ∨ S n + ; Cof2 : M n2 t i n ηq n + −−−−−− → M n2 r i M −− → C n + M −− → M n + t → M n + r ; Cof3 : S n ∨ M n2 t ( r ,ηq n + ) −−−−−−−− → S n i n −− → C n + M −− → S n + ∨ M n + t → S n + ; Cof4 : S n + (cid:16) i n η2 t (cid:17) −−−−−− → M n2 r ∨ S n + M −− → C n + n + −−−− → S n + → M n + r ∨ S n + ; Cof5 : C n +
1r 2 t q n + −−−−− → S n + n + −−− → C n + C −− → C n + → S n + ; Cof6 : S n i n r −−− → C n + C −− → C n + n + −−−− → S n + → C ( n + ) ,t . Lemma 2.1. (1) For the canonical inclusions i n + k from S n + k into C n + or C n + , there holda relation: i n η = t · i n + . (2) For the canonical projections i n + k from C n + or C n + onto S n + k , there hold a relation: ηq n + = r · q n + . PENGCHENG LI
Recall that for a cofiber sequence X f − → Y i − → Z q − → ΣX Σf −− → ΣY of A -spaces and for any W ∈ A , n ≥ , there are exact sequence of abelian groups: [ ΣY, W ] ( Σf ) ∗ −−− → [ ΣX, W ] q ∗ −− → [ Z, W ] i ∗ − → [ Y, W ] f ∗ − → [ X, W ] , (2.1) [ W, X ] f ∗ − → [ W, Y ] i ∗ − → [ W, Z ] q ∗ −− → [ W, ΣX ] ( Σf ) ∗ −−− → [ W, ΣY ] (2.2)2.2. Spanier-Whitehead duality.Definition 2.2 ([22]) . Let n > k + .(1) Let X ∈ A kn , a (Spanier-Whitehead) ( + k ) -duality is a CW-complex X d ∈ A kn with a map µ : X d ∧ X → S + k , which induces the following natural isomorphisms for any Z, W ∈ A kn : D µ : [ Z, W ∧ X d ] ∧ X −− → [ Z ∧ X, W ∧ X d ∧ X ] ( ∧ µ ) d −−−−− → [ Z ∧ X, W ∧ S + k ] D µ ( f ) = ( ∧ µ )( f ∧ ); µ D : [ Z, X ∧ W ] X d ∧ −−− → [ X d ∧ Z, X d ∧ X ∧ W ] ( µ ∧ ) d −−−−− → [ X d ∧ Z, S + k ∧ W ] µ D ( g ) = ( µ ∧ )( ∧ g ) . (2) There is a contravariant isomorphism functor D = D + k : A kn → A kn , called the Spanier-Whiteheadduality functor, such that DX = X d , Df = f d ∈ [ Y d , X d ] for any f ∈ [ X, Y ] , where f d is characterized by the homotopy equality: µ X ( f d ∧ X ) ≃ µ Y ( Y d ∧ f ) , where µ X , µ Y are the ( + k ) -duality maps of X, Y , respectively.
Lemma 2.3 (Proposition 2.1 of [25]) . The Spanier-Whitehead duality functor D satisfies the followingproperties:(1) DD = id : X dd = X, f dd = f .(2) ( X ∧ Y ) d = Y d ∧ X d and ( f ∧ g ) d = g d ∧ f d .(3) ( X ∨ Y ) d = X d ∨ Y d .(4) [ X, Y ] ∼ = [ Y d ∧ X, S + k ] ∼ = [ Y d , X d ] . Example 2.4 (Example 2.2 of [25]) . Let D = D + , then DS n + i = S + − i , DM n2 r = M n + r , DC n + = C n + , DC n + = C n + , DC n + = C n + . Example 2.5.
The canonical homotopy equivalence S n + − i ∧ S n + i → S + is a Spanier-Whitehead ( + ) -duality map; D + η n + = η n : S n + → S n .2.3. The Steenrod squares.
For basic knowledge of the Steenrod squares Sq : H k ( X ; Z /2 ) → H k + ( X ; Z /2 ) we refer to Chapter 4.L of [13]. The following lemma is useful. Lemma 2.6.
Let X be a wedge sum of elementary Chang-complexes. The Steenrod square Sq : H n ( X ; Z /2 ) → H n + ( X ; Z /2 ) is a natural isomorphism with respect to X . Homotopy Classification of Maps between A -complexes In this section we give a new proof of Baues’s group structures of the sets of homotopy classes ofmaps between A ( n ≥ ) -complexes and choose their explicit generators. Proposition 3.1.
The following groups determine exactly all non-trivial groups [ X, Y ] with X = M n + ip r ( i =
0, 1 ) : ( i ) [ M np r , S n + ] ∼ = Z /p r h q n + i . ( ii ) [ M np r , M np t ] ∼ = Z /p min ( r,t ) h B n ( χ ) i , [ M np r , M n + t ] ∼ = Z /p min ( r,t ) h i n + q n + i . ( iii ) [ M n + t , C n + ] ∼ = Z /p t h ˜ ζq n + i , [ M n + t , C n + ] ∼ = Z /p t h i η ˜ ζq n + i . OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 9 Proof. ( i ) , ( ii ) refer to [7]. ( iii ) Let X = C n + , C n + . Since π n + ( X ) = , there is an exact sequenceof groups: π n + ( X ) p r − → π n + ( X ) q ∗ n + −−− → [ M n + r , X ] i ∗ n + −−− → π n + ( X ) = Then ( iii ) follows from the group structures of π n + ( X ) . (cid:3) Proof of Table 1.
The groups structures and generators above the double lines in Table 1 wereproved by Brown and Copeland [8] and the notations of the generators are due to Baues and Hennes[7]. The proofs of items in Table 1 under the double lines can be divided into the following twopropositions.
Proposition 3.2.
The stable homotopy groups π sn + i ( X )( i ≤ ) are given by the table: S n S n + S n + C n + Z i n Z ˜ ζC n + Z i n Z /2 t + i n + Z /2 i n + ηC n + Z /2 r i n Z i η ˜ ζ ⊕ Z /2 i M ˜ η r C n + Z /2 r i n Z /2 t + i n + Z /2 i n + η ⊕ Z /2 i M ˜ η r where ˜ ζ : S n + → C n + satisfies q n + ˜ ζ ≃ · n + . (3.1) Proof.
Applying the exact functor π sn + i (−)( i =
0, 1 ) to the cofiber sequences Cof1 for the four el-ementary Chang-complexes, it’s easy to get the first columns in the above table. We only give theproofs in the case i = here. C n + : There is an exact sequence of groups: π sn + ( S n + ) η ∗ −− → ∼ = π sn + ( S n ) ( i n ) ∗ −−− → π sn + ( C n + ) ( q n + ) ∗ −−−−−− → π sn + ( S n + ) η ∗ −− → π sn + ( S n + ) . Hence π sn + ( C n + ) ( q n + ) ∗ −−−−−− → ∼ = Z h · n + i . Let ˜ ζ ∈ π sn + ( C n + ) satisfies (3.1), then the proof of π sn + ( C n + ) ∼ = Z h ˜ ζ i is completed. C n + : Applying π sn + (−) to Cof2 for C n + , there is an exact sequence: π sn + ( M n2 t ) ( ηq n + ) ∗ −−−−−−− → π sn + ( S n ) ( i n ) ∗ −−−− → π sn + ( C n + ) ( q M ) ∗ −−−− → π sn + ( M n + t ) −− → π sn + ( S n + ) . Since π sn + ( M n2 ) ∼ = Z /4 h ˜ η i ; π sn + ( M n2 t ) ∼ = Z /2 h i n η i ⊕ Z /2 h ˜ η t i ( t > 1 ) and q n + ˜ η t ≃ η , by somecomputations we get Coker ( ηq n + ) ∗ = ; Ker ( ηq n + ) ∗ = π sn + ( M n + t ) ∼ = Z /2 h i n + η i . By the composition i n + = q M i n + , , π sn + ( C n + ) ∼ = Z /2 h i n + η i is proved. C n + : Applying π sn + (−) to Cof2 for C n + , there is an exact sequence: π sn + ( S n + ) ( i n η ) ∗ −−−− → π sn + ( M n2 r ) ( i M ) ∗ −−−− → π sn + ( C n + ) ( q n + ) ∗ −−−−−− → π sn + ( S n + ) −− → π sn + ( M n + r ) . Since i n η = ˜ η , the above sequence turns to the following splitting short exact sequence: → Z /2 h ˜ η r i ( i M ) ∗ −−−− → π sn + ( C n + ) ( q n + ) ∗ −−−−−− → Z h · n + i → n + : Applying π sn + (−) to Cof4 for C n + , there is an exact sequence: π sn + ( S n + ) ( inη2t ) ∗ −−−−− → π sn + ( M n2 r ∨ S n + ) ( i M ,i n + ) ∗ −−−−−−−− → π sn + ( C n + ) → We have Coker (cid:18) i n η2 t (cid:19) ∗ = Z /2 h { ˜ η r } i ⊕ Z /2 h η i , where { ˜ η r } denotes the coset class represented by ˜ η r . Then the group structure and generators of π sn + ( C n + ) follows. (cid:3) Note.
By Lemma 2.3 ( ) , π n + is ( X ) = [ X, S n + i ] ∼ = π sn + − i ( X d ) , where X is an elementary Chang-complex; the generators can be similarly obtained. Proposition 3.3.
Let X be an elementary A -complex, the groups [ M n2 r , X ] and [ M n2 t , X ] are given bythe table below. M n2 r M n + t C n + Z /2 t ˜ ζq n + C n + ′ Z /2 m ′′ i n + q n + Z /2 n ′′ ⊕ Z /2 ˜ ξB n + ( χ ) , i n + ηq n + C n + ′ Z /2 l i M B n ( χ ) Z /2 t ⊕ Z /2i η ˜ ζq n + , i M ˜ η r ′ q n + C n + ′ r ′ Z /2 l ⊕ Z /2 m ′′ i M B n ( χ ) , i n + q n + Z /2 n ′′ ⊕ Z /2 ⊕ Z /2i C ˜ ξB n + ( χ ) , i M ˜ η r ′ q n + , i n + ηq n + where l = min ( r, r ′ ) , m ′′ = min ( r, t ′ + ) , n ′′ = min ( t.t ′ ) ; the map ˜ ξ : M n + t + → C n + satisfies ˜ ξi n + = i n + . (3.2) Proof. C n + : Applying the exact functor [− , C n + ] to the cofiber sequences for M n2 r , M n + t , respec-tively, there are exact sequence of groups: = [ S n + , C n + ] q ∗ n + −−− → [ M n2 r , C n + ] i ∗ n − → [ S n , C n + ] r − → [ S n , C n + ];[ S n + , C n + ] t − → [ S n + , C n + ] q ∗ n + −−− → [ M n + t , C n + ] → [ S n + , C n + ] = Then it follows that [ M n2 r , C n + ] = [ M n + t , C n + ] ∼ = Z /2 t h ˜ ζq n + i . C n + ′ : Applying [ M n2 r , −] to Cof3 for C n + ′ , there is an exact sequence: [ M n2 r , C n + ] ( t ′ q n + ) ∗ −−−−−−−− → [ M n2 r , S n + ] ( i n + ) ∗ −−−−− → [ M n2 r , C n + ′ ] → [ M n2 r , C n + ] = where [ M n2 r , C n + ] ∼ = Z /2 r h Σ − ( ˜ ζq n + ) i . By 3.1 we get [ M n2 r , C n + ′ ] ∼ = Z /2 min ( r,t ′ + ) h i n + q n + i . For the group [ M n + t , C n + ′ ] , [ S n + , C n + ′ ] ∼ = Z /2 t ′ + h i n + i implies that there exists an extension˜ ξ : M n + t ′ + − → C n + ′ such that ˜ ξi n + = i n + . Let f = (cid:0) i n + ( Σ ¯ η ) , ˜ ξ (cid:1) : M n + ∨ M n + t ′ + → C n + ′ , then one checks that f ♯ : π sn + i ( M n + ∨ M n + t ′ + ) → π sn + i ( C n + ′ ) is an isomorphism if i = and anepimorphism if i = . Consider the following commutative diagram induced by f , where rows are exact OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 11 sequences: π sn + ( M n + ∨ M n + t ′ + ) t (cid:15) (cid:15) f ∗ / / / / π sn + ( C n + ′ ) t (cid:15) (cid:15) π sn + ( M n + ∨ M n + t ′ + ) q ∗ n + (cid:15) (cid:15) f ∗ / / / / π sn + ( C n + ′ ) q ∗ n + (cid:15) (cid:15) [ M n + t , M n + ∨ M n + t ′ + ] f ∗ / / i ∗ n + (cid:15) (cid:15) [ M n + t , C n + ′ ] i ∗ n + (cid:15) (cid:15) π sn + ( M n + ∨ M n + t ′ + ) t (cid:15) (cid:15) f ∗ ∼ = / / π sn + ( C n + ′ ) t (cid:15) (cid:15) π sn + ( M n + ∨ M n + t ′ + ) f ∗ ∼ = / / π sn + ( C n + ′ ) After computing cokernels and kernels of the homorphisms t , we have the following commutativediagram with exact rows and columns: / / Z /2 h i n + + i n + η i (cid:15) (cid:15) (cid:15) (cid:15) q ∗ n + / / Ker ( f ∗ ) (cid:15) (cid:15) (cid:15) (cid:15) / / (cid:15) (cid:15) / / Z /2 h i n + i ⊕ Z /2 h i n + η i f ∗ (cid:15) (cid:15) (cid:15) (cid:15) q ∗ n + / / [ M n + t , M n + ∨ M n + t ′ + ] f ∗ (cid:15) (cid:15) (cid:15) (cid:15) / / K / / ∼ = (cid:15) (cid:15) / / Z /2 h i n + η i q ∗ n + / / [ M n + t , C n + ′ ] / / K / / where K = Ker (cid:18) Z /2 t ′ + h i n + i t −− → Z /2 t ′ + h i n + i (cid:19) , the first row is obtained by the snake lemma.By the group structure [ M n + t , M n + ∨ M n + t ′ + ] ∼ = Z /2 h i n + q n + i ⊕ Z /2 min ( t,t ′ + ) h B n + ( χ ) i ⊕ Z /2 h i n + ηq n + i ∼ = Z /2 min ( t,t ′ + ) h B n + ( χ ) i ⊕ Z /2 h i n + ηq n + i⊕ Z /2 h i n + q n + + i n + ηq n + i and the exactness of the middle column, we get an isomorphism: [ M n + t , C n + ′ ] ∼ = Z /2 min ( t,t ′ + ) h ˜ ξB n + ( χ ) i ⊕ Z /2 h i n + ηq n + i .C n + ′ : Applying [ M n2 r , −] to Cof2 for C n + ′ , there is an exact sequence: [ M n2 r , S n + ] i n η −−− → [ M n2 r , M n2 r ′ ] i M −− → [ M n2 r , C n + ′ ] → [ M n2 r , S n + ] = where [ M n2 r , S n + ] ∼ = Z /2 r h q n + i . By the group structure and generators of [ M n2 r , M n2 r ′ ] (if r = r ′ = , · M = i n ηq n + ), we get [ M n2 r , C n + ′ ] ∼ = Z /2 min ( r,r ′ ) h i M B n ( χ ) i . For [ M n + t , C n + ′ ] , there is an exact sequence: [ S n + , C n + ′ ] t −− → [ S n + , C n + ′ ] q ∗ n + −−−− → [ M n + t , C n + ′ ] → [ S n + , C n + ′ ] = Then the isomorphism immediately follows: Z h i η ˜ ζ i ⊕ Z /2 h i M ˜ η r ′ ih t · i η ˜ ζ i q ∗ n + −−−− → ∼ = [ M n + t , C n + ′ ] .C n + ′ r ′ : Let g = ( i M , i C ) : M n2 r ′ ∨ C n + ′ → C n + ′ r ′ , then g ♯ : π sn + i ( M n2 r ′ ∨ C n + ′ ) → π sn + i ( C n + ′ r ′ ) is an epimorphism for i =
0, 1 . Consider the following commutative diagram induced by g with exactcolumns: π sn + ( M n2 r ′ ∨ C n + ′ ) r (cid:15) (cid:15) g ∗ / / / / π sn + ( C n + ′ r ′ ) r (cid:15) (cid:15) π sn + ( M n2 r ′ ∨ C n + ′ ) q ∗ n + (cid:15) (cid:15) g ∗ / / / / π sn + ( C n + ′ r ′ ) q ∗ n + (cid:15) (cid:15) [ M n2 r , M n2 r ′ ∨ C n + ′ ] g ∗ / / i ∗ n (cid:15) (cid:15) [ M n2 r , C n + ′ r ′ ] i ∗ n (cid:15) (cid:15) π sn ( M n2 r ′ ∨ C n + ′ ) r (cid:15) (cid:15) g ∗ / / / / π sn ( C n + ′ r ′ ) r (cid:15) (cid:15) π sn ( M n2 r ′ ∨ C n + ′ ) g ∗ / / / / π sn ( C n + ′ r ′ ) By similar arguments as that in the proof of [ M n + t , C n + ′ ] , there is a reduced commutative diagramof exact sequences: / / Z /2 h ( i n η, 2 min ( r − ′ ) { i n + } ) i q ∗ n + / / (cid:15) (cid:15) Ker ( g ∗ ) (cid:15) (cid:15) / / / / (cid:15) (cid:15) / / Z /2 h i n η i ⊕ Z /2 min ( r,t ′ + ) h { i n + } i q ∗ n + / / g ∗ (cid:15) (cid:15) (cid:15) (cid:15) [ M n2 r , M n2 r ′ ∨ C n + ′ ] g ∗ (cid:15) (cid:15) i ∗ n / / K ∼ = (cid:15) (cid:15) / / / / Z /2 min ( r,t ′ + ) h { i n + } i q ∗ n + / / [ M n2 r , C n + ′ r ′ ] i ∗ n / / K / / where K = Ker (cid:0) π sn ( M n2 r ′ ∨ C n + ′ ) r − → π sn ( M n2 r ′ ∨ C n + ′ ); K = Ker (cid:0) π sn ( C n + ′ r ′ ) r − → π sn ( C n + ′ r ′ ) . By the snake lemma, g ∗ : [ M n2 r , M n2 r ′ ∨ C n + ′ ] → [ M n2 r , C n + ′ r ′ ] is an epimorphism. Since [ M n2 r , M n2 r ′ ∨ C n + ′ ] ∼ = Z /2 min ( r,r ′ ) h B n ( χ ) i ⊕ Z /2 h i n ηq n + i ⊕ Z /2 min ( r,t ′ + ) h i n + q n + i , we get [ M n2 r , C n + ′ r ′ ] ∼ = Z /2 min ( r,r ′ ) h i M B n ( χ ) i ⊕ Z /2 min ( r,t ′ + ) h i n + q n + i . OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 13 For the group [ M n + t , C n + ′ r ′ ] , consider the extension diagram: M n + ∨ C n + ′ ( i M ˜ η r ′ ,i C ) * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ S n + ∨ S n + ( i n + ,i n + ) O O ( i M ˜ η r ′ ,i n + ) / / C n + ′ r ′ where i M ˜ η r ′ is an extension of i M ˜ η r ′ over M n + . One can check the induced isomorphisms h = ( i M ˜ η r ′ , i C ) ∗ : π j ( M n + ∨ C n + ′ ) ∼ = −− → π j ( C n + ′ r ′ ) , j = n +
1, n + Similarly, by the commutative diagram of exact sequences induced by h and the five-lemma, there isan isomorphism [ M n + t , C n + ′ r ′ ] ( i M ˜ η r ′ ,i C ) ∗ ← −−−−−−−− ∼ = [ M n + t , M n + ∨ C n + ′ ] Let n ′′ = min ( t, t ′ + ) , then we get [ M n + t , C n + ′ r ′ ] ∼ = Z /2 h i M ˜ η r ′ q n + i ⊕ Z /2 n ′′ h i C ˜ ξB n + ( χ ) i ⊕ Z /2 h i n + ηq n + i . (cid:3) Remark . ˜ ξ ∈ [ M n + t + , C n + ] simultaneously satisfies the following relation equalities after choos-ing suitably: ˜ ξi n + ≃ i n + , q η ˜ ξ ≃ ˜ ζq n + , q n + ˜ ξ ≃ n + , q M t ˜ ξ ≃ B n + ( χ ); (3.3)Dually, there exist a map ¯ ξ ∈ [ C n + , M n2 t + ] simultaneously satisfies the relation equalities: q n + ¯ ξ ≃ q n + , ¯ ξi η ≃ i n ¯ ζ, ¯ ξi n ≃ n , ¯ ξi M t ≃ B n ( χ ) . (3.4) Proof.
We only prove (3.3) here and omit the similar proof of (3.4). Consider the diagram withcofibration rows: S n + t + / / Σ − ˜ ζ (cid:15) (cid:15) S n + n + / / M n + t + q n + / / ˜ ξ (cid:15) (cid:15) ✤✤✤ S n + ˜ ζ (cid:15) (cid:15) C n +
1η 2 t q n + / / S n + n + / / C n + η / / C n + By (3.1), the first square is (homotopy) commutative, which implies that there exist a map ˜ ξ filling inthe right two commutative squares: ˜ ξi n + = i n + , q η ˜ ξ = ˜ ζq n + . By (3.1), we have q n + ˜ ζ = q n + q η ˜ ξ = q n + ˜ ζq n + = n + . For the last relation equality, by the group structure and generators of [ M n + t + , M n + t ] , we may put q M t ˜ ξ = x · B n + ( χ ) + y · i n + ηq n + for some x ∈ Z /2 t , y ∈ Z /2 . By composing q M on both sides of the equality from the left, togetherwith (1.2), we get n + = q n + ˜ ξ = q n + q M t ˜ ξ = x · q n + B n + ( χ ) = · q n + . Thus x = .If y = , the proof is done; otherwise, substituting ˜ ξ by ˜ ξ + i n + ηq n + , then q M t ˜ ξ = B n + ( χ ) holds. One can check that the new ˜ ξ satisfies all the relation equalities discussed above and the proofis completed. (cid:3) Proof of Table 2.
By Lemma 2.3 ( ) , it’s easy to get the group structures of [ X, Y ] with X anelementary Chang-complex and Y a sphere or an elementary Moore spaces in A ; the generators canbe obtained by similar arguments or by the Spanier-Whitehead duality. It suffices to show the itemsbelow the double lines in Table 2, which are proved by the following proposition. Proposition 3.5.
The groups and generators of maps between elementary Chang-complexes are exactlydetermiend by the following groups. [ C n + , C n + ] ∼ = Z h η i ⊕ Z h i n ¯ ζ i ∼ = Z h η i ⊕ Z h ˜ ζq n + i , where ¯ ζ ∈ [ C n + , S n ] and ˜ ζ ∈ [ S n + , C n + ] satisfy the relations: q n + ˜ ζ = · n + , ¯ ζi n = · n ; i n ¯ ζ + ˜ ζq n + = · η . (3.5) [ C n + , C n + ′ ] ∼ = Z h i n ¯ ζ i , [ C n + , C n + ′ ] ∼ = Z h i η i ⊕ Z /2 r ′ h i n ¯ ζ i , [ C n + , C n + ′ r ′ ] ∼ = Z /2 r ′ h i n ¯ ζ i . [ C n + , C n + ′ ] ∼ = (cid:12) Z /2 t ′ + h ˜ ξB n + ( χ ) q M i ⊕ Z h i n ¯ ζq η i t > t ′ ; Z /2 t h ˜ ξB n + ( χ ) q M i ⊕ Z h ˜ κ i t ≤ t ′ , where ˜ κ satisfies the relations: ˜ κi n = i n , ˜ κ = C ( t = t ′ ); q ′ M ˜ κ = B n + ( χ ) q M , q η ˜ κ = q η . (3.6) i n ¯ ζq η = · ˜ κ − ˜ ξB n + ( χ ) q M . (3.7) [ C n + , C n + ′ ] ∼ = (cid:12) Z /2 t + h i η q η i ⊕ Z /2 r ′ h i n ¯ ζq η i , t ≥ r ′ ; Z /2 r ′ + h i η q η i ⊕ Z /2 t h i η ˜ ζq n + i , t < r ′ ; . [ C n + , C n + ′ r ′ ] ∼ = Z /2 r ′ + h i C ˜ κ i ⊕ Z /2 t h i C ˜ ξB n + ( χ ) q M i t ′ ≥ t < r ′ ; Z /2 t + h i C ˜ κ i ⊕ Z /2 r ′ h i n ¯ ζq η i t ′ ≥ t ≥ r ′ ; Z /2 t ′ + h i C ˜ ξB n + ( χ ) q M i ⊕ Z /2 r ′ h i n ¯ ζq η i t ′ < t . . [ C n + , C n + ′ ] ∼ = Z /2 min ( r,t ′ ) h i n + q n + i . [ C n + , C n + ′ r ′ ] ∼ = Z /2 min ( r + ′ ) h i M B n ( χ ) ¯ ξ i ⊕ Z /2 min ( r,t ′ )+ h i n + q n + i . Let m = min ( r, t ′ ) , l ′ = min ( r +
1, r ′ ) , n ′′ = min ( t, t ′ + ) , j = max ( t, r ′ ) , then [ C n + , C n + ′ r ′ ] ∼ = Z /2 l ′ h i M B n ( χ ) ¯ ξq C i ⊕ Z /2 n ′′ h i C ˜ ξB n + ( χ ) q M i , r ′ > r ∨ t ′ < t ; Z /2 j + h i C ˜ κ i ⊕ (cid:12) Z /2 r ′ h i M B n ( χ ) ¯ ξq C i t ≥ r ′ ; Z /2 t h i C ˜ ξB n + ( χ ) q M i t < r ′ . r ′ ≤ r ∧ t ≤ t ′ ⊕ Z /2 m + h i n + q n + i . where i C ˜ κ satisfies the relations: i C ˜ κi C = i C ˜ κ, i C ˜ κi M = i ′ M B n ( χ ) , q ′ M i C ˜ κ = B n + ( χ ) q M . (3.8) Proof. Applying [ C n + , −] to Cof1 for C n + , there is an exact sequence: → [ C n + , S n ] ( i n ) ∗ −−−− → [ C n + , C n + ] ( q n + ) ∗ −−−−−− → [ C n + , S n + ] → Since [ C n + , S n ] ∼ = Z h ¯ ζ i , [ C n + , S n + ] ∼ = Z h q n + i , where ¯ ζ satisfies ¯ ζi n = · n , the above exactsequence splits. Hence the group [ C n + , C n + ] is proved. The other generator set follows from therelation equality (see [2]): i n ¯ ζ + ˜ ζq n + = · η .2. Applying [ C n + , −] to Cof1 s for C n + ′ , C n + ′ , C n + ′ r ′ , respectively, together with Table 1,the proof of ( ) is easy and omitted here. Applying [− , C n + ′ ] to Cof2 for C n + , there is an exact sequence: [ S n + , C n + ′ ] ( ηq n + ) ∗ −−−−−−− → [ M n + t , C n + ′ ] q ∗ M −− → [ C n + , C n + ′ ] i ∗ n −− → [ S n , C n + ′ ] ( ηq n + ) ∗ −−−−−−− → [ M n2 t , C n + ′ ] OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 15 Then we get Z /2 t ′ + h i n + i ( ηq n + ) ∗ −−−−−−− → Z /2 n ′′ h ˜ ξB n + ( χ ) i ⊕ Z /2 h i n + ηq n + i q ∗ M −− → [ C n + , C n + ′ ] i ∗ n −− → Z h i n i ( ηq n + ) ∗ −−−−−−− → Z /2 n ′′ h i n + q n + i , where n ′′ = min ( t, t ′ + ) . By Lemma 2.1, i n η = t ′ · i n + ∈ [ S n + , C n + ′ ] , we have the following two splitting short exactsequences: t > t ′ : → Z /2 t ′ + h ˜ ξB n + ( χ ) i q ∗ M −− → [ C n + , C n + ′ ] i ∗ n −− → Z h · i n i → ; t ≤ t ′ : → Z /2 t h ˜ ξB n + ( χ ) i q ∗ M −− → [ C n + , C n + ′ ] i ∗ n −− → Z h i n i → If t > t ′ , note that i ∗ n ( i n ¯ ζq η ) = · i n , we have [ C n + , C n + ′ ] ∼ = Z /2 t ′ + h ˜ ξB n + ( χ ) q M i ⊕ Z h i n ¯ ζq η i . If t ≤ t ′ , let ˜ κ ∈ [ C n + , C n + ′ ] satisfies˜ κi n = i n , ˜ κ = C ( t = t ′ ) , then we get [ C n + , C n + ′ ] ∼ = Z /2 t h ˜ ξB n + ( χ ) q M i ⊕ Z h ˜ κ i . For the relation equalities (3.6), (3.7), consider the following commutative diagram with cofibrationrows: M n2 t B n ( χ ) (cid:15) (cid:15) ηq n + / / S n i n / / n (cid:15) (cid:15) C n + M / / ˜ κ (cid:15) (cid:15) ✤✤✤ M n + t ΣB n ( χ )= B n + ( χ ) (cid:15) (cid:15) M n2 t ′ ηq ′ n + / / S n i ′ n / / C n + ′ q ′ M / / M n + t ′ Hence there exists a map ˜ κ satisfies˜ κi n = i n , q ′ M ˜ κ = B n + ( χ ) q M . By the group [ C n + , C n + ] , we can set q η ˜ κ = x · q η + y · ˜ ζq n + for some x ∈ Z , y ∈ Z /2 t . By composing q n + on the both sides of the equality from the left, we have ( x + ) · q n + q n + ˜ κ = q n + ˜ κ = q n + q ′ M ˜ κ = q n + B n + ( χ ) q M = q n + , hence x + = . Then composing i n on the both sides of the equality from the right, we have i n = x · i n and hence x =
1, y = . Thus q η ˜ κ = q η is proved.Since i n ¯ ζq η i n = i n ¯ ζi n = n ∈ [ S n , C n + ′ ] , ˜ κi n = i n , we may put ( R x ) i n ¯ ζq η = x · ˜ ξB n + ( χ ) q M + · ˜ κ , x ∈ Z /2 t . By composing q η on the both sides of the equality ( R x ) from the left, we have · q η − ˜ ζq n + = q η i n ¯ ζq η = i n ¯ ζq η , by ( 3.5) = x · q η ˜ ξB n + ( χ ) q M + · q η ˜ κ = x · ˜ ζq n + B n + ( χ ) q M + · q η ˜ κ, by (3.3) = x · ˜ ζq n + + · q η , by (3.6) and the composition: C n + n + ( ( q M / / M n + t B n + ( χ ) / / M n + t ′ + q n + / / S n + ˜ ζ / / C n + Thus x = − , (3.7) is proved. Applying [− , C n + ′ ] to Cof3 for C n + , there is an exact sequence: [ S n + , C n + ′ ] ( t q n + ) ∗ −−−−−−− → [ C n + , C n + ′ ] q ∗ η −− → [ C n + , C n + ′ ] → [ S n + , C n + ′ ] = where [ S n + , C n + ′ ] ∼ = Z h i η ˜ ζ i ⊕ Z /2 h i M ˜ η r ′ i , [ C n + , C n + ′ ] ∼ = Z h i η i ⊕ Z /2 r ′ h i n ¯ ζ i .If t ≥ r ′ , by ( 3.5), t · i η ˜ ζq n + = ( t + · i η , 0 ) , it follows that [ C n + ′ , C n + ′ ] ∼ = Z /2 t + h i η q η i ⊕ Z /2 r ′ h i n ¯ ζq η i . If t < r ′ , consider the exact sequence induced by Cof3 for C n + ′ : [ C n + , S n ] i n r ′ −−−− → [ C n + , C n + ] i η −− → [ C n + , C n + ′ ] → hence r ′ · i n ¯ ζq η ≃ ( r ′ + · q η , 0 ) and we get [ C n + , C n + ′ ] ∼ = Z /2 r ′ + h i η q η i ⊕ Z /2 t h i η ˜ ζq n + i .5. Applying [ C n + , −] to Cof6 for C n + ′ r ′ , there is an exact sequence: ( ∗ ) [ C n + , S n ] ( i n r ′ ) ∗ −−−−−− → [ C n + , C n + ′ ] ( i C ) ∗ −−−− → [ C n + , C n + ′ r ′ ] → If t ′ < t , ( ∗ ) can be reduced to Z h ¯ ζq η i i n r ′ −−−− → Z /2 t ′ + h ˜ ξB n + ( χ ) q M i ⊕ Z h i n ¯ ζq η i i C −− → [ C n + , C n + ′ r ′ ] → Hence [ C n + , C n + ′ r ′ ] ∼ = Z /2 t ′ + h i C ˜ ξB n ( χ ) q M i ⊕ Z /2 r ′ h i n ¯ ζq η i . If t ′ ≥ t , applying [ C n + , −] to Cof5 for C n + ′ r ′ , there is an exact sequence: ( ∗∗ ) → [ C n + , C n + ′ r ′ ] ( q C ) ∗ −−−− → [ C n + , C n + ′ ] ( t ′ q n + ) ∗ −−−−−−−− → [ C n + , S n + ] ∼ = Z /2 t h q n + i . In this case, the homomorphism ( t ′ q n + ) ∗ = , hence ( ⋆ ) [ C n + , C n + ′ r ′ ] q Cn + ′ −−−−− → ∼ = [ C n + , C n + ′ ] . Then the sequence ( ∗ ) turns to ( ∗ ′ ) Z h ¯ ζq η i ( i n r ′ ) ∗ −−−−−− → Z /2 t h ˜ ξB n + ( χ ) q M i ⊕ Z h ˜ κ i ( i C ) ∗ −−−− → [ C n + , C n + ′ r ′ ] → If r ′ ≥ t ≤ t ′ , then [ C n + , C n + ′ r ′ ] ∼ = Z /2 t h i C ˜ ξB n + ( χ ) q M i ⊕ Z /2 r ′ + h i C ˜ κ i . If r ′ ≤ t ≤ t ′ , then [ C n + , C n + ′ r ′ ] i C ← −− ∼ = Z /2 t h ˜ ξB n + ( χ ) q M i ⊕ Z h ˜ κ ih r ′ (− ˜ ξB n + ( χ ) q M , 2 ˜ κ ) i =: G ∼ = Z /2 t + h { ˜ κ } i ⊕ Z /2 r ′ h X i , where X = (− ˜ ξB n + ( χ ) q M , 2 · ˜ κ ) = i n ¯ ζq η , the group structure and generators G is proved as follows.Let t ≥ r ′ , consider the homomorphism φ : Z /2 t h a i ⊕ Z h b i − → Z /2 r ′ h ˜ a i ⊕ Z /2 t + h ˜ b i , φ ( a ) = ˜ a + ( b ) = ˜ b. then φ ( a − ) = ˜ a , φ is surjective andKer ( φ ) = { ( x · a, y · b ) ∈ Z /2 t ⊕ Z ; x ≡ ( mod r ′ ) , 2x + y ≡ ( mod t + ) } = h (− r ′ a, 2 r ′ + b ) . i It follows that there is an isomorphism ¯ φ : G = Z /2 t h a i⊕ Z h b ih (− r ′ a,2 r ′ + b ) i ∼ = − → Z /2 r ′ h ˜ a i ⊕ Z /2 t + h ˜ b i such that¯ φ ( { a − } ) = ˜ a, ¯ φ ( { b } ) = ˜ b, OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 17 where { a } is the coset class with a presentative a . Let a = ˜ ξB n + ( χ ) q M , b = ˜ κ , then we get G ∼ = Z /2 r ′ h − X i ⊕ Z /2 t + h { ˜ κ } i ∼ = Z /2 r ′ h X i ⊕ Z /2 t + h { ˜ κ } i . Applying [− , C n + ′ ] to Cof1 for C n + , there is an exact sequence: π n + ( C n + ′ ) ( ∗ ) −−− → π n + ( C n + ′ ) ⊕ π n + ( C n + ′ ) ( q ∗ n + ,q ∗ n + ) −−−−−−−−− → [ C n + , C n + ′ ] → Thus Z /2 t ′ + h i n + i ⊕ Z /2 h i n + η ih ( r i n + , i n + η ) i ∼ = Z /2 m + h { i n + } i q ∗ n + −−−− → ∼ = [ C n + , C n + ′ ] , where m = min ( r, t ′ ) , { i n + } is the coset class with a presentative i n + . Consider the commutative diagram with exact columns induced by the map h = ( i M , i C ) : M n2 r ′ ∨ C n + ′ → C n + ′ r ′ : π n + ( M n2 r ′ ∨ C n + ′ ) h ∗ / / ( ∗ ) (cid:15) (cid:15) π n + ( C n + ′ r ′ )( ∗ ) (cid:15) (cid:15) π n + ( M n2 r ′ ∨ C n + ′ ) ⊕ π n + ( M n2 r ′ ∨ C n + ′ ) h ∗ / / ( q ∗ n + ,q ∗ n + ) (cid:15) (cid:15) π n + ( C n + ′ r ′ ) ⊕ π n + ( C n + ′ r ′ ) ( q ∗ n + ,q ∗ n + ) (cid:15) (cid:15) [ C n + , M n2 r ′ ∨ C n + ′ ] h ∗ / / i ∗ n (cid:15) (cid:15) [ C n + , C n + ′ r ′ ] i ∗ n (cid:15) (cid:15) π n ( M n2 r ′ ∨ C n + ′ ) h ∗ / / ( ∗ ) (cid:15) (cid:15) π n ( C n + ′ r ′ )( ∗ ) (cid:15) (cid:15) π n ( M n2 r ′ ∨ C n + ′ ) ⊕ π n + ( M n2 r ′ ∨ C n + ′ ) h ∗ / / π n ( C n + ′ r ′ ) ⊕ π n + ( C n + ′ r ′ ) After computing cokernels and kernels of the four homorphisms (cid:0) r η ∗ (cid:1) , we get the commutative diagramof exact sequences: Z /2 m + h { i n + } i ⊕ Z /2 h i M ˜ η r ′ i / / ( q ∗ n + ,q ∗ n + ) / / [ C n + , C n + ′ r ′ ] / / / / K Z /2 m + h { i n + } i ⊕ Z /2 h { ˜ η r ′ } i ⊕ Z /2 h i n η i / / / / O O O O [ C n + , M n2 r ′ ∨ C n + ′ ] / / / / ( i M ,i C ) ∗ O O O O K Z /2 h ( i n η, ε · t ′ { i n + } ) i O O O O / / q ∗ n + / / Ker ( i M , i C ) ∗ O O O O / / O O O O where ε = (cid:12) if r ≥ t ′ if r < t ′ ; K = Ker (cid:18) Z /2 r ′ h i n i ( r ,η ∗ ) −−−−− → Z /2 r ′ h i n i ⊕ Z /2 t ′ + h i n + i (cid:19) .By the smake lemma, Ker ( i M , i C ) ∗ = Z /2 h ( i n ηq n + , 2 m i n + q n + i . Then by the exactness of the middle column and the group [ C n + , M n2 r ′ ∨ C n + ′ ] ∼ = Z /2 l ′ h B n ( χ ) ¯ ξ i ⊕ Z /2 m + h i n + q n + i ⊕ Z /2 h i n ηq n + i , we get [ C n + , C n + ′ r ′ ] ∼ = Z /2 l ′ h i M B n ( χ ) ¯ ξ i ⊕ Z /2 m + h i n + q n + i , where l ′ = min ( r +
1, r ′ ) , m = min ( r, t ′ ) . We only give the proof of the case r ′ > 1 and omit the similar discussion in the case r ′ = here.Firstly we have ( r ′ > 1 ): [ S n ∨ S n + , C n + ′ r ′ ] ∼ = Z /2 r ′ h i n i ⊕ Z /2 t ′ + h i n + i ;[ S n + ∨ S n + , C n + ′ r ′ ] ∼ = Z /2 t ′ + h i n + i ⊕ Z /2 h i M ˜ η r ′ i ⊕ Z /2 h i n + η i ;[ S n ∨ S n + , M n2 r ′ ∨ C n + ′ ] ∼ = Z /2 r ′ h i n i ⊕ Z h i n i ⊕ Z /2 h i n η i ⊕ Z /2 t ′ + h i n + i ;[ S n + ∨ S n + , M n2 r ′ ∨ C n + ′ ] ∼ = Z /2 h i n η i ⊕ Z /2 t ′ + h i n + i⊕ Z /2 h ˜ η r ′ i ⊕ Z /2 h i n η i ⊕ Z /2 h i n + η i . Consider the commutative diagram with exact columns induced by h = ( i M , i C ) : M n2 r ′ ∨ C n + ′ ֒ → C n + ′ r ′ : [ S n + ∨ S n + , M n2 r ′ ∨ C n + ′ ] h ∗ / / (
2r η ∗ ) (cid:15) (cid:15) [ S n + ∨ S n + , C n + ′ r ′ ](
2r η ∗ ) (cid:15) (cid:15) [ S n + ∨ S n + , M n2 r ′ ∨ C n + ′ ] h ∗ / / ( qn + + ) ∗ (cid:15) (cid:15) [ S n + ∨ S n + , C n + ′ r ′ ]( qn + + ) ∗ (cid:15) (cid:15) [ C n + , M n2 r ′ ∨ C n + ′ ] h ∗ / / ( i n ,i n + ) ∗ (cid:15) (cid:15) [ C n + , C n + ′ r ′ ] ( i n ,i n + ) ∗ (cid:15) (cid:15) [ S n ∨ S n + , M n2 r ′ ∨ C n + ′ ] h ∗ / / (
2r η ∗ ) (cid:15) (cid:15) [ S n ∨ S n + , M n2 r ′ ∨ C n + ′ ](
2r η ∗ ) (cid:15) (cid:15) [ S n ∨ S n + , M n2 r ′ ∨ C n + ′ ] h ∗ / / [ S n ∨ S n + , M n2 r ′ ∨ C n + ′ ] Then we get a commutative diagram with exact rows:Coker (cid:0) r η ∗ t (cid:1) / / ( qn + + ) ∗ / / [ C n + , C n + ′ r ′ ] ( i n ,i n + ) ∗ / / / / Ker (cid:0) r η ∗ t (cid:1) =: K ′ Coker (cid:0) r η ∗ t (cid:1) / / ( qn + + ) ∗ / / h ∗ O O [ C n + , M n2 r ′ ∨ C n + ′ ] h ∗ O O ( i n ,i n + ) ∗ / / / / Ker (cid:0) r η ∗ t (cid:1) =: K h ∗ | K O O Case 1: r ′ > r ∨ t ′ < t : By computations, K ′ = { ( x · i n , y · i n + ); ( r x, 2 t ′ x + t y ) = (
0, 0 ) ∈ Z /2 r ′ ⊕ Z /2 t ′ + , x ∈ Z /2 r ′ , y ∈ Z /2 t ′ + } = Z /2 r h r ′ − r i n i ⊕ Z /2 t ′ + h i n + ) i , t > t ′ , r < r ′ Z /2 r h r ′ − r i n i ⊕ Z /2 t h t ′ + − t i n + i , t ≤ t ′ , r < r ′ Z /2 r ′ − h n i ⊕ Z /2 t ′ + h i n + ) i , t > t ′ , r ≥ r ′ K = Z /2 h i n η i ⊕ Z /2 r h r ′ − r i n i ⊕ Z /2 t ′ + h i n + i , t > t ′ , r < r ′ Z /2 r h r ′ − r i n i ⊕ Z /2 t h t ′ + − t i n + i , t ≤ t ′ , r < r ′ Z /2 r ′ − h n i ⊕ Z /2 t ′ + h i n + i , t > t ′ , r ≥ r ′ Z /2 r ′ − h n i ⊕ Z /2 t h t ′ + − t i n + i , t ≤ t ′ , r ≥ r ′ . OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 19 Since i n η ≃ t ′ i n + ∈ [ S n + , C n + ′ r ′ ] , in the case r ′ > r ∨ t ′ < t , the homomorphism h ∗ | K is surjectiveand there holds: Ker ( h ∗ | K ) = Z /2 h ( i n η, 2 t ′ i n + ) i . After computing the cokernels and applying the snake lemma, we have the following commutative ofexact sequences: ( ⋆ ) Z /2 t ′ + h i n + i⊕ Z /2 h i n + η ih ( r · i n + ,i n + η ) i ⊕ Z /2 h i M ˜ η r ′ i / / q ∗ n + / / [ C n + , C n + ′ r ′ ] i ∗ n + / / / / K ′ Z /2 h i n η i ⊕ Z /2 t ′ + h i n + i⊕ Z /2 h i n + η ih ( r · i n + ,i n + η ) i ⊕ Z /2 h ˜ η r ′ i / / q ∗ n + / / h ∗ O O O O [ C n + , M n2 r ′ ∨ C n + ′ ] i ∗ n + / / / / ( i M ,i C ) ∗ O O O O K h ∗ | K O O O O Z /2 h ( i n η, 2 t ′ ε · { i n + } ) i / / q ∗ n + / / O O O O Ker ( i M , i C ) ∗ i ∗ n + ∗ n + / / / / O O O O Ker ( g ) O O O O where ε = if r ≥ t ′ ; otherwise ε = . By the group [ C n + , M n2 r ′ ∨ C n + ′ ] ∼ = Z /2 l ′ h B n ( χ ) ¯ ξq C i ⊕ Z /2 h i n ηq n + i ⊕ Z /2 h i n ¯ η t q M i⊕ Z /2 n ′′ h ˜ ξB n + ( χ ) q M i ⊕ Z /2 m + h i n + q n + i we check that the subgroup Z /2 h i n η i is isomorphically mapped by q ∗ n + onto a direct summand,which implies the splitting exactness of the third rows of ( ⋆ ) . By (3.4), ˜ ξi n + ≃ i n + , we have (cid:0) i n ¯ η t q M + min ( t ′ ,t − ) ˜ ξB n + ( χ ) q M (cid:1) i n + = i n ¯ η t i n + + min ( t ′ ,t − ) ˜ ξB n + ( χ ) i n + = i n η + t ′ i n + . Hence Ker ( i M , i C ) ∗ = Z /2 h ( i n ηq n + , 2 t ′ ε · i n + q n + ) i ⊕ Z /2 h ( i n ¯ η t q M , 2 min ( t ′ ,t − ) ˜ ξB n + ( χ ) q M ) i .Combining the exactness of the middle column of ( ⋆ ) , we see that if r ′ > r ∨ t ′ < t , [ C n + , C n + ′ r ′ ] ∼ = Z /2 l ′ h i M B n ( χ ) ¯ ξq C i ⊕ Z /2 n ′′ h i C ˜ ξB n + ( χ ) q M i ⊕ Z /2 m + h i n + q n + i . Case 2: r ′ ≤ r ∧ t ≤ t ′ . By computations, K ′ = { ( x · i n , y · i n + ); t ′ x + t y ≡ ( mod t ′ + ) , x ∈ Z /2 r ′ , y ∈ Z /2 t ′ + } K = Z /2 r ′ − h · i n i ⊕ Z /2 t h t ′ + − t · i n + i ⊕ Z /2 h i n η i . Coker ( h ∗ | K ) = { ( x · i n , y · i n + ); x ∈ Z /2 r ′ , y ∈ Z /2 t ′ + , 2 t ′ x + t y ≡ ( mod t ′ + ) } h · i n i ⊕ h t ′ + − t · i n + i ∼ = Z /2 h ( i n , 2 t ′ − t · i n + ) i Ker ( h ∗ | K ) = Z /2 h ( i n η, 2 t ′ i n + ) i . There is a commutative diagram of exact sequences: ( ⋆⋆ ) / / Z /2 ∼ = / / ( h ∗ | K ) Z /2 m + h { i n + } i ⊕ Z /2 h i M ˜ η r ′ i / / q ∗ n + / / O O O O [ C n + , C n + ′ r ′ ] i ∗ n + / / / / O O O O K ′ O O O O Z /2 h i n η i ⊕ Z /2 m + h { i n + } i ⊕ Z /2 h ˜ η r ′ i / / q ∗ n + / / h ∗ O O O O [ C n + , M n2 r ′ ∨ C n + ′ ] i ∗ n + / / / / ( i M ,i C ) ∗ O O K h ∗ | K O O Z /2 h ( i n η, ε · t ′ { i n + } ) i / / q ∗ n + O O O O Ker ( i M , i C ) ∗ i ∗ n + / / / / O O O O ker ( g ) O O O O where Ker ( i M , i C ) ∗ = Z /2 h ( i n ηq n + , 2 t ′ ε · i n + q n + ) i ⊕ Z /2 h ( i n ¯ η t q M , 2 t − ˜ ξB n + ( χ ) q M ) i . By theexactness of the middle column and the group of [ C n + , M n2 r ′ ∨ C n + ′ ] , we have the followingextension: → Z /2 r ′ ⊕ Z /2 t ⊕ Z /2 m + h i n + q n + i − → [ C n + , C n + ′ r ′ ] → Z /2 → Let E = [ C n + , C n + ′ r ′ ] , A = Z /2 r ′ ⊕ Z /2 t ⊕ Z /2 m + , then group cohomology correspondingto the extension E is H ( Z /2, A ) ∼ = H ( L ∞ ; A ) ∼ = Ext ( Z /2, A ) ∼ = A/2A ∼ = Z /2 ⊕ Z /2 ⊕ Z /2 , where L ∞ = S ∞ / Z /2 is the lens space. It follows that E has four possible group structures: A ⊕ Z /2, Z /2 m + ⊕ Z /2 r ′ ⊕ Z /2 t , Z /2 m + ⊕ Z /2 r ′ + ⊕ Z /2 t , Z /2 m + ⊕ Z /2 r ′ ⊕ Z /2 t + . We claim that Z /2 m + h i n + q n + i is a direct summand of E . If E ∼ = Z /2 m + ⊕ Z /2 r ′ ⊕ Z /2 t , then i n + q n + = , z ∈ E . Let z ′ ∈ K ′ be the projection image of z , then ′ = . By the group structure of K ′ , z ′ = ( x · i n , · i n + ) satisfies x ≡ ( mod r ′ − ) . On the other hand, by the commutativity of the right-up square of ( ⋆⋆ ) and thegroup of Coker ( h ∗ | K ) , we see that z ′ = ( x · i n , y · i n + ) satisfies x ≡ ( mod ) , contradiction. Thusthe claim is proved.We now determine the group structure and generators of [ C n + , C n + ′ r ′ ] as follows. Applying [− , C n + ′ r ′ ] to Cof6 for C n + , there is an exact sequence: ( ♦ ) [ S n + , C n + ′ r ′ ] q ∗ n + −−−− → [ C n + , C n + ′ r ′ ] i ∗ C − → [ C n + , C n + ′ r ′ ] r · i ∗ n −−−− → [ S n , C n + ′ r ′ ] If r ′ ≤ r ∧ t ≤ t ′ , the homomorphisms r · i ∗ n = , ( ♦ ) is equivalent to ( ♦ ′ ) → Z /2 m + h { i n + } i q ∗ n + −−−− → [ C n + , C n + ′ r ′ ] i ∗ C −− → [ C n + , C n + ′ r ′ ] → [ C n + , C n + ′ r ′ ] ∼ = (cid:14) Z /2 r ′ + h i C ˜ κ i ⊕ Z /2 t h i C ˜ ξB n + ( χ ) q M i t ′ ≥ t < r ′ Z /2 t + h i C ˜ κ i ⊕ Z /2 r ′ h i n ¯ ζq η i t ′ ≥ t ≥ r ′ .Since Z /2 m + h i n + q n + i is a direct summand, the exact sequence ( ♦ ′ ) actually splits. Let j = max ( t, r ′ ) , k = min ( t, r ′ ) , i C ˜ κ ∈ [ C n + , C n + ′ r ′ ] satisfies i C ˜ κi C = i C ˜ κ, we get [ C n + , C n + ′ r ′ ] ∼ = Z /2 m + h i n + q n + i ⊕ Z /2 j + h i C ˜ κ i ⊕ Z /2 k . If t ′ ≥ t < r ′ , note that i C ˜ ξB n + ( χ ) q M ∈ [ C n + , C n + ′ r ′ ] satisfies i C ˜ ξB n + ( χ ) q M i C ≃ i C ˜ ξB n + ( χ ) q M , the direct summand Z /2 k = Z /2 t is generated by the map i C ˜ ξB n + ( χ ) q M . OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 21 If t ′ ≥ t ≥ r ′ , q C i C ∈ [ C n + , C n + ] ∼ = Z /2 max ( r,t )+ h i η q η i ⊕ Z /2 min ( r,t ) h Y i , where Y = (cid:12) i n ¯ ζq η , t ≥ ri η ˜ ζq n + , t < r . We clain that q C i C = i η q η . (3.9)We may put q C i C = a · i η q η + b · Y . If t ≥ r , By composing i n on the both sides of the equality fromthe left, we have i n = ( a + ) · i n . By composing ˜ ζq n + on the both sides of the equality from the right, we have˜ ζq n + ≃ a · ˜ ζq n + + Thus a =
1, b = , (3.9) is proved. The proof of (3.9) in the case t < r is similar and the proof of theclain is completed.By the duality of (3.3), i M B n ( χ ) ¯ ξq C ∈ [ C n + , C n + ′ r ′ ] satisfies i M B n ( χ ) ¯ ξq C i C ≃ i M B n ( χ ) ¯ ξi η q η = i M B n ( χ ) i n ¯ ζq η = i n ¯ ζq η , Thus Z /2 k = Z /2 r ′ is generated by i M B n ( χ ) ¯ ξq C in this case.To sum up, if r ′ ≤ r ∧ t ≤ t ′ , let j = max ( t, r ′ ) , we get [ C n + , C n + ′ r ′ ] ∼ = Z /2 m + h i n + q n + i ⊕ Z /2 j + h i C ˜ κ i ⊕ (cid:12) Z /2 r ′ h i M B n ( χ ) ¯ ξq C i t ≥ r ′ ; Z /2 t h i C ˜ ξB n + ( χ ) q M i t < r ′ . For the relations (3.8), consider the diagram with exact rows: M n2 t i n ηq n + / / B n ( χ ) (cid:15) (cid:15) M n2 r B n ( χ ) (cid:15) (cid:15) i M / / C n + C ˜ κ (cid:15) (cid:15) q M / / M n + t B n + ( χ ) (cid:15) (cid:15) M n2 t ′ i n ηq n + / / M n2 r ′ i ′ M / / C n + ′ r ′ q ′ M / / M n + t ′ By the commutativity of the first square, it suffices to show that i C ˜ κi M = i ′ M B n ( χ ) . We may put i C ˜ κi M ≃ x · i ′ M B n ( χ ) + y · i n + q n + for some x ∈ Z /2 r ′ , y ∈ Z /2 min ( r,t ′ + ) . Note that i C ˜ κi M i n = ˜ κi n = i n , we get x = . If y = , theproof is done; otherwise, substituting i C ˜ κ by i C ˜ κ − y · i n + q n + , then i C ˜ κi M = i ′ M B n ( χ ) follows since i n + q n + i C = . (cid:3) There is an alternative method to get generators of [ X, Y ] by combing Baues’s group structuresand the following lemma. Lemma 3.6.
Given an exact sequence of groups: ( ES ) → A f −− → B g −− → C → or A f −− → B g −− → C If the following conditions hold: ( i ) A, B, C be finitely generated abelian groups is an exact sequence, ( ii ) B ∼ = A ⊕ C ,then the exact sequence ( ES ) splits.Proof. The condition ( ii ) implies that Ker ( f ) = and im ( g ) = C . It suffices to show the lemma inthe case where A, B, C are Z -modules of finite lengths.Note that the exact sequence ( ES ) splits if and only if the sequence → Hom ( C, A ) g ∗ −− → Hom ( B, A ) f ∗ −− → Hom ( A, A ) is right-exact; i.e., im ( f ∗ ) = Hom ( A, A ) . Let l ( M ) be the length of an R -module M . For the exact sequence ( ES ) , there holds l ( B ) = l ( A ) + l ( C ) . Then l ( Hom ( B, A )) = l ( Hom ( C, A )) + l ( Hom ( A, A ));= l ( Hom ( C, A )) + l ( im ( f ∗ )) and hence l ( im ( f ∗ )) = l ( Hom ( A, A )) , im ( f ∗ ) = Hom ( A, A ) . Thus the exact sequence ( ES ) splits. (cid:3) Example 3.7.
There are splitting short exact sequences: ( ) t ′ ≥ t : → Z /2 m + h i n + i q ∗ n + −−− → [ C n + , C n + ′ r ′ ] i ∗ C − → [ C n + , C n + ′ r ′ ] → ;( ) t ′ < t : → [ C n + , , C n + ′ r ′ ] q ∗ C −− → [ C n + , C n + ′ r ′ ] i ∗ n + −−− → Z /2 t ′ + h i n + i → Applications
The decomposability of smash products of A -complexes. Using the generators in Ta-ble 1 and Table 2, we re-study the decomposability of the homotopy types of the smash products ofelementary A -complexes, which was firstly completed by Zhu and Pan [25] in 2017. Let n ≥ . Lemma 4.1 ( Lemma 6.2.1 of [15]) . Let X f − → Y i − → Z q − → ΣX be a cofiber sequences in FCW ∗ , in which Y is simply-connected. The following statements are equivalent: ( i ) f is null homotopic. ( ii ) there exists a splitting retraction r : Z → Y such that ri = Y . ( iii ) there exists a splitting section s : ΣX → Z such that qs = ΣX . ( iv ) h = ( i, s ) : Y ∨ ΣX → Z is a homotopy equivalence. Definition 4.2.
An cofiber sequence ( X f − → ) Y i − → Z q − → ΣX is called splitting if any one of the fourstatements in Lemma 4.1 holds.A homotopy analog of Lemma 3.6 is the following: Proposition 4.3.
Let X f − → Y g − → Z be a cofiber sequence in the stable homotopy category such that Y ≃ X ∨ Z , then the sequence is a splitting cofiber sequence.Proof. In the stable homotopy category the cofiber sequence X f − → Y g − → Z induces an exact sequenceof groups: [ Z, X ] f ∗ − → [ Z, Y ] g ∗ − → [ Z, Z ] . Since Y ≃ X ∨ Z , [ Z, Y ] ∼ = [ Z, X ] ⊕ [ Z, Z ] ; then by Lemma 3.6 the above exact sequence splits.Hence there is a map ϑ ∈ [ Z, Y ] satisfying g ∗ ( ϑ ) = gϑ = Z . The proposition then follows by Lemma4.1. (cid:3) The following two theorems are direct corollaries of Proposition 4.3 and Theorem 1.1 of [25]:
Theorem 4.4. M n2 u ∧ M n2 r , M n2 u ∧ C ′ , C ′ ∈ { C n + , C n + , C n + } There are splitting cofiber sequecens for M n2 u ∧ M n2 r : ( i ) u ≥ r and u > 1 : S n ∧ M n2 r i n ∧ −−−− → M n2 u ∧ M n2 r q n + ∧ −−−−−− → S n + ∧ M n2 r . ( ii ) r ≥ u and r > 1 : M n2 u ∧ S n 1 ∧ i n −−−− → M n2 u ∧ M n2 r ∧ q n + −−−−−− → M n2 u ∧ S n + . ( iii ) u = r = : M n2 ∧ M n2 ≃ C n + ∧ S n is given by the commutative diagram of cofiber sequences: M n2 ∧ S n 2 / / M n2 ∧ S n 1 ∧ i n / / M n2 ∧ M n2 1 ∧ q n + / / ¯ β (cid:15) (cid:15) M n2 ∧ S n + M n2 ∧ S n iηq ∧ / / M n2 ∧ S n i M ∧ / / C n + ∧ S n ¯ α J J q M ∧ / / M n + ∧ S n where ¯ α is the homotopy inverse of ¯ β ; i.e., ¯ β ¯ α ≃ id , ¯ α ¯ β ≃ id . OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 23 If r, t ≥ u , there is a splitting cofiber sequences for M n2 u ∧ C ′ : ( i ) M n2 u ∧ S n + ∧ i n + −−−−− → M n2 u ∧ C n + ∧ q η −−−− → M n2 u ∧ C n + . ( ii ) M n2 u ∧ C n +
2η 1 ∧ i η −−−− → M n2 u ∧ C n +
2r 1 ∧ q n + −−−−−− → M n2 u ∧ S n + . ( iii ) M n2 u ∧ C n + ∧ i C −−−− → M n2 u ∧ C n + ∧ q n + −−−−−− → M n2 u ∧ S n + or M n2 u ∧ S n + ∧ i n + −−−−− → M n2 u ∧ C n + ∧ q C −−−− → M n2 u ∧ C n + . M n2 u ∧ C n + ≃ M n2 u ∧ C n + ∨ M n2 u ∧ S n + , r ≥ u > tM n2 u ∧ S n + ∨ M n2 u ∧ C n + ∨ M n2 u ∧ S n + , r, t ≥ uM n2 u ∧ S n + ∨ M n2 u ∧ C n + , t ≥ u > r3. If u > r, t , M n2 u ∧ C n + and M n2 u ∧ C n + are indecomposable (see Theorem 1.1 of [25]); there isa splitting cofiber sequence for M n2 u ∧ C n + : S n ∧ C n + n ∧ −−−− → M n2 u ∧ C n + n + ∧ −−−−−− → S n + ∧ C n + . Theorem 4.5.
Let C ′ ∈ { C n + ′ , C n + ′ , C n + ′ r ′ } , then the decomposability of C ′ ∧ C n + are listedas follows.
1. C n + ′ ∧ C n + ≃ S n + ∧ C n + ∨ C n + ∧ C n + , if t ′ ≥ t, r ; S n + ∧ C n + ∨ C n + ∧ C n + , if t ′ = r < t indecomposable , otherwise . If t ′ ≥ t, r , there is a splitting cofiber sequence: S n + ∧ C n + n + ∧ C −−−−−− → C n + ′ ∧ C n + η ∧ C −−−−− → C n + ∧ C n + .2. C n + ′ ∧ C n + ≃ C n + ∧ C n + ∨ S n + ∧ C n + , if r ′ ≥ r, t ; S n + ∧ C n + ∨ C n + ∧ C n + , if r ′ = t < r indecomposable , otherwise . If r ′ ≥ r, t , there is a splitting cofiber sequence: C n + ∧ C n + η ∧ C −−−−− → C n + ′ ∧ C n + n + ∧ C −−−−−−− → S n + ∧ C n + .3. C n + ′ r ′ ∧ C n + ≃ S n + ∧ C n + ∨ C n + ′ ∧ C n + , t = t ′ ≥ r ′ > r ∨ r = t ′ > r ′ > t ; S n + ∧ C n + ∨ C n + ′ ∧ C n + , t = r ′ > t ′ > r ∨ r = r ′ > t ′ > t ; C n + ′ r ′ ∧ S n + ∨ C n + ′ r ′ ∧ C n + , t ≥ r, r ′ , t ′ ∧ other cases ; C n + ′ r ′ ∧ S n + ∨ C n + ′ r ′ ∧ C n + , r ≥ r ′ , t ′ , r > t ∧ other cases . Especially, ( i ) C n + ∧ C n + ≃ S n + ∧ C n + ∨ S n + ∧ C n + ∨ C n + ∧ C n + ,where m = min ( r, t ) and the corresponding splitting cofiber sequences are t ≥ r : C n + ∧ C n + C ∧ C −−−−− → C n + ∧ C n + n + ∧ C −−−−−−− → S n + ∧ C n + . t < r : C n + ∧ C n + C ∧ i C −−−−− → C n + ∧ C n + C ∧ q n + −−−−−−− → C n + ∧ S n + . ( ii ) C n + ∧ C n + ≃ (cid:14) S n + ∧ C n + ∨ C n + ∧ C n + , t ≥ rS n + ∧ C n + ∨ C n + ∧ C n + , t < r . The corresponding splitting cofiber sequences are as follows. t ≥ r : S n + ∧ C n + n + ∧ C −−−−−− → C n + ∧ C n + C ∧ C −−−−− → C n + ∧ C n + .t < r : C n + ∧ C n + C ∧ C −−−−− → C n + ∧ C n + n + ∧ C −−−−−−− → S n + ∧ C n + . Remark . For the reasons of the length of the article, in Theorem 4.5 we don’t give the splittingcofiber sequences for the decomposable smash products C n + ∧ C n + ≃ S n + ∧ C n + ∨ C n + ∧ C n + ( t < r ) and its dual (interchanging t and r ): C n + ∧ C n + ≃ S n + ∧ C n + ∨ C n + ∧ C n + ( r < t ) . We remark without proof here that there is a splitting cofiber sequence for C n + ∧ C n + if t < r : C n + ∧ C n + ( C ∧ i C ) e T −−−−−−− → C n + ∧ C n + t −− → C n + ∧ S n + , where(1) e T : C n + ∧ C n + → C n + ∧ C n + is a lift of the switching map T ( C η , C ) : C n + ∧ C n + → C n + ∧ C n + satisfying the relation ( C ∧ q η ) e T = T ( C η , C )( η ∧ q C ) , (4.1)(2) D t : C n + ∧ C n + → C n + ∧ S n + is an extension of the Spanier-Whitehead ( + ) -dualitymap D t : C n + ∧ C n + → S + satisfying the relation D t ( C ∧ i C ) = ( i n + ∧ n + ) D t . (4.2)By Lemma 4.1, there is an alternative proof of Theorem 4.5 is to show some special maps are nullhomotopic. Actually, this was the first method I adopted to prove Theorem 4.4, 4.5 without usingTheorem 1.1 of [25]. The detailed proof can be separated into the following three lemmas: Lemma4.7,4.8,4.9. Lemma 4.7.
Let m = min ( r, t ) . ( ) [ C n + , ΣC n + ] ∼ = Z /2 r h ( Σi n ) q n + i ⊕ Z /2 t h ( Σi n + ) q n + i . ( ) [ C n + , ΣC n + ] ∼ = Z /2 r h ( Σi n ) q n + i ⊕ Z /2 t h ( Σi n + ) q n + i ⊕ Z /2 m h ( Σi M ) B n + q M i .Proof. ( ) Applying the exact functor [− , ΣC n + ] to Cof1 for C n + , there is an exact sequence: [ S n + ∨ S n + , ΣC n + ] r η0 2 t ∗ −−−−−−−−−− → [ S n + ∨ S n + , ΣC n + ] ( q ∗ n + ,q ∗ n + ) −−−−−−−−− → [ C n + , ΣC n + ] ( i ∗ n ,i ∗ n + ) −−−−−−− → [ S n ∨ S n + , ΣC n + ] −− → [ S n ∨ S n + , ΣC n + ] . After substituting the group structures ofthe hoomotopy groups of C n + , we get an exact sequence: Z h Σi n i ⊕ Z /2 t + h Σi n + i r η0 2 t ∗ / / Z h Σi n i ⊕ Z /2 t + h Σi n + i ( q ∗ n + ,q ∗ n + ) / / / / [ C n + , ΣC n + ] . Hence by Lemma 2.1,Coker (cid:18) r η0 2 t (cid:19) ∗ = Z h Σi n i ⊕ Z /2 t + h Σi n + ih ( r · Σi n , 2 t · Σi n + ) i ⊕ h (
0, 2 t · Σi n + ) i ∼ = Z /2 r h { Σi n } i ⊕ Z /2 t h { Σi n + } i . ( ) Since [ S n , ΣC n + ] = , applying [− , ΣC n + ] to Cof3 for C n + , there is an exact sequence: [ S n + , ΣC n + ] ( r ,ηq n + ) ∗ −−−−−−−−− → [ S n + ∨ M n + t , ΣC n + ] q ∗ ¯ M −− → [ C n + , ΣC n + ] → Then by Table 1 and Lemma 2.1,Coker ( r , ηq n + ) ∗ = Z /2 r h Σi n i ⊕ Z /2 t h Σ ( i n + ) q n + ) i ⊕ Z /2 m h ( Σi M ) B n + ( χ ) ih ( r · Σi n , ( Σi n η ) q n + i = h (
0, 2 t · ( Σi n + ) q n + ) i = ∼ = Z /2 r h Σi n i ⊕ Z /2 t h Σ ( i n + ) q n + ) i ⊕ Z /2 m h ( Σi M ) B n + ( χ ) i , OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 25 the lemma is proved. (cid:3) Lemma 4.8. ( ) q n + ∧ C ∈ [ C n + ∧ C n + , S n + ∧ C n + ] is of order max ( r,t ) ; ( ) i n ∧ C ∈ [ S n ∧ C n + , C n + ∧ C n + ] is of order max ( r,t ) .Proof. Consider the following exact sequence: [ S n + ∧ C, S n + ∧ C ] ( η n + ∧ C ) ∗ −−−−−−−−− → [ S n + ∧ C, S n + ∧ C ] ( q n + ∧ C ) ∗ −−−−−−−−− → [ C n + ∧ C, S n + ∧ C ] → · · · Recall that [ S n + ∧ C, S n + ∧ C ] ∼ = Z /2 m + h n + ∧ i n + q n + i ⊕ Z /2 j + h n + ∧ C i ⊕ Z /2 m h n + ∧ ω tr i , where m = min ( r, t ) , j = max ( r, t ) , ω tr = (cid:12) i M B n ( χ ) ¯ ξq C , t ≥ r ; i C ˜ ξB n + ( χ ) q M , t < r. By the monomorphism Coker ( η n + ∧ C ) ∗ / / ( q n + ∧ ) ∗ / / [ C n + ∧ C, S n + ∧ C ] , q n + ∧ C is theimage of the coset class { n + ∧ C } = n + ∧ C + Im ( η n + ∧ C ) ∗ , hence ( Eq.0 ) | q n + ∧ C | = |{ n + ∧ C }| . By Lemma 2.1, i n η n = t i n + , η n + q n + = r q n + ,we have the following relation equalities: ( i ) ( n + ∧ ( Σi n ) q n + )( η n + ∧ C ) = n + ∧ i n η n q n + = t · n + ∧ i n + q n + (= if t > r );( ii ) ( n + ∧ ( Σi n + ) q n + )( η n + ∧ C ) = n + ∧ i n + η n + q n + = r · n + ∧ i n + q n + (= if r > t );( iii ) ( n + ∧ ( Σi M ) B n + ( χ ) q M )( η n + ∧ C ) = (cid:0) n + ∧ i M B n ( χ ) (cid:1) ( η n + ∧ M )( n + ∧ q M )= n + ∧ i M B n ( χ ) i n ¯ η t q M + n + ∧ i M B n ( χ ) ˜ η t q n + where the last equality is due to Toda(Lemma 7.2 of [2]): η n + ∧ M t = n + ∧ i n ¯ η t + n + ∧ ˜ η t q n + . Omitting the notation Σ n + = n + ∧ below, we haveCoker ( η n + ∧ C ) ∗ ∼ = Z /2 m h { i n + q n + } i ⊕ Z /2 j + h C i ⊕ Z /2 m h ω tr ih i M B n ( χ ) i n ¯ η t q M + i M B n ( χ ) ˜ η t q n + i . By the properties of B n ( χ ) , it’s easy to compute that i M B n ( χ ) i n ¯ η t q M = (cid:12) i n ¯ η t q M , if t ≥ r0, if t < r ; i M B n ( χ ) ˜ η t q n + = (cid:12) if t > ri M ˜ η r q n + , if t ≤ r . We now determine i n ¯ η t q M and i M ˜ η r q n + by relations in Proposition 1.2 as follows.Case 1: t ≥ r . Put ( Eq.1 ) i n ¯ η t q M = x · i n + q n + + y · C + z · i M B n ( χ ) ¯ ξq C for some x ∈ Z /2 r + , y ∈ Z /2 t + , z ∈ Z /2 r . By composing i n + on the both sides of ( Eq.1 ) from theright, we have i n η n = t i n + = y · i n + ∈ π n + ( C ) ∼ = Z /2 t + h i n + i , hence y = t .By composing i M on the both sides of ( Eq.1 ) from the right, we have = x · i n + q n + + t i M + · i M ∈ [ M n2 r , C ] ∼ = Z /2 r h i M i ⊕ Z /2 r h i n + q n + i , Hence x ≡ ( mod r ) and ≡ ( mod r ) . By composing q n + on the both sides of ( Eq.1 ) from the left, we have = t · q n + + · q n + ∈ [ C, S n + ] ∼ = Z /2 r + h q n + i . Hence z ≡ ( mod r ) if t > r and z ≡ r − ( mod r ) if t = r .To sum up, we get ( ∗ ) i n ¯ η t q M = x · i n + q n + + t · C + r − ε · i M B n ( χ ) ¯ ξq C , where x ≡ ( mod r ) , ε = if t > r ; ε = if t = r .If t = r , put ( Eq.2 ) i M ˜ η r q n + = x ′ · i n + q n + + y ′ · C + z ′ · i M B n ( χ ) ¯ ξq C for some x ′ , y ′ ∈ Z /2 r + , z ∈ Z /2 r . By composing i n + on the both sides of ( Eq.2 ) from the right,we have = y ′ · i n + ∈ π n + ( C ) ∼ = Z /2 r + h i n + i , hence y ′ = .By composing q n + on the both sides of ( Eq.2 ) from the right, we have η n + q n + = r · q n + = ′ · q n + ∈ [ C, S n + ] ∼ = Z /2 r + h q n + i , hence z ′ = r − .By composing i M on the both sides of ( Eq.2 ) from the right, we have = x ′ · i n + q n + + ′ · i M ∈ [ M n2 r , C ] ∼ = Z /2 r h i M i ⊕ Z /2 r h i n + q n + i , hence x ′ ≡ ( mod r ) .Thus if t = r , we get a relation: ( ∗∗ ) i M ˜ η r q n + = x ′ · i n + q n + + r − · i M B n ( χ ) ¯ ξq C , where x ′ ≡ ( mod r ) .Combining with relations ( ∗ ) , ( ∗∗ ) , we conclude that if t > r ,Coker ( η n + ∧ C ) ∗ ∼ = Z /2 r h { i n + q n + } i ⊕ Z /2 t + h C i ⊕ Z /2 r h i M B n ( χ ) ¯ ξq C ih ( x · i n + q n + , 2 t · C , 0 ) = (
0, 2 t · C , 0 ) i ∼ = Z /2 r h { i n + q n + } i ⊕ Z /2 t h { C } i ⊕ Z /2 r h i M B n ( χ ) ¯ ξq C i ; if t = r , Coker ( η n + ∧ C ) ∗ ∼ = Z /2 r h { i n + q n + } i ⊕ Z /2 r + h C i ⊕ Z /2 r h i M B n ( χ ) ¯ ξq C ih (
0, 2 r · C , 0 ) i ∼ = Z /2 r h { i n + q n + } i ⊕ Z /2 r h { C } i ⊕ Z /2 r h i M B n ( χ ) ¯ ξq C i . Case 2: t < r . We may put ( Eq.3 ) i M ˜ η r q n + = x ′′ · i n + q n + + y ′′ · C + z ′′ · i C ˜ ξB n + ( χ ) q M for some x ′′ ∈ Z /2 t + , y ′′ ∈ Z /2 r + , z ′′ ∈ Z /2 t . By composing i M on the both sides of ( Eq.3 ) fromthe right, we have = x ′′ · i n + q n + + y ′′ · i M ∈ [ M n2 r , C ] ∼ = Z /2 t + h i n + q n + i ⊕ Z /2 r h i M i , hence x ′′ ≡ ( mod t + ) , y ′′ ≡ ( mod r ) .By composing i n + on the both sides of ( Eq.3 ) from the right, we have = ( y ′′ + ′′ ) · i n + , hence y ′′ + ′′ ≡ ( mod t + ) .Thus x ′′ =
0, y ′′ = r , z ′′ = and we get i M ˜ η r q n + = r · C . In this case,Coker ( η n + ∧ C ) ∗ ∼ = Z /2 t h { i n + q n + } i ⊕ Z /2 r + h C i ⊕ Z /2 t h i C ˜ ξB n + ( χ ) q M ih (
0, 2 r · C , 0 ) i ∼ = Z /2 t h { i n + q n + } i ⊕ Z /2 r h { C } i ⊕ Z /2 t h i C ˜ ξB n + ( χ ) q M i . OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 27 By the above discussion, we obtain the following isomorphismCoker ( η n + ∧ C ) ∗ ∼ = Z /2 min ( r,t ) h { n + ∧ i n + q n + } i ⊕ Z /2 max ( r,t ) h { n + ∧ C } i⊕ Z /2 min ( r,t ) h n + ∧ ω tr i . By ( Eq.0 ) , the proof of ( ) is completed. ( ) By Spanier-Whitehead duality, [ S n ∧ C n + , C n + ∧ C n + ] ∼ = [ C n + ∧ C n + , C n + ∧ S n + ] ∼ = [ C n + ∧ C n + , S n + ∧ C n + ] . By Definition 2.2, q n + : C n + → S n + the ( + ) -duality map of i n : S n → C n + . It then followsfrom ( ) that | i n ∧ C | = | C ∧ q n + | = | q n + ∧ C | = max ( r,t ) . (cid:3) Lemma 4.9. ( ) q n + ∧ C ∈ [ C n + ′ ∧ C n + , S n + ∧ C n + ] is of order min ( r ′ , max ( r,t )) . ( ) i n ∧ C ∈ [ S n ∧ C n + , C n + ′ ∧ C n + ] is of order min ( t ′ , max ( r,t )) .Proof. ( ) Let C = C n + , consider the following exact sequence: [ S n + ∧ C, S n + ∧ C ] ⊕ [ S n + ∧ C, S n + ∧ C ] (( η n + ∧ C ) ∗ ,2 r ′ ) (cid:15) (cid:15) [ S n + ∧ C, S n + ∧ C ] ( q n + ∧ C ) ∗ (cid:15) (cid:15) [ C n + ′ ∧ C, S n + ∧ C ] Hence q n + ∧ C is the image of the coset { n + ∧ C } in Coker (( η n + ∧ C ) ∗ , 2 r ′ ) . By the proof ofLemma 4.8, we compute thatCoker (( η n + ∧ ) ∗ , 2 r ′ ) ∼ = Z /2 min ( r,t ) ⊕ Z /2 max ( r,t ) h { C } i ⊕ Z /2 min ( r,t ) r ′ [ h (
1, 0, 0 ) i ⊕ h (
0, 1, 0 ) i ⊕ h (
0, 0, 1 ) i ] . Thus | q n + ∧ C | = |{ n + ∧ C }| = min ( r ′ , max ( r,t )) .The proof of ( ) is totally parallel to ( ) . (cid:3) The following results about C n + are an essential application of Theorem 4.5: Theorem 4.10 (Theorem 1.1 of [24]) . Let n ≥ . ΩΣC n + ≃ Y j ΩΣC k j ( n + )+ × (some other space) , where < k < · · · is a sequence of odd integers such that k j is not a multiple of any k i else foreach j . Corollary 4.11 (Corollary 1.2 of [24]) . Let p ≥ be an odd integer with p ≥ k − n + + , then group π sk ( C n + ) is a direct summand of π k +( p − )( n + ) ( C n + ) for n ≥ . The groups of self-homotopy equivalences of A -complexes. Using generators in Table 1and Table 2 and relations in Proposition 1.2), it’s easy to get the following lemma(or refer to [6]):
Lemma 4.12.
The groups E ( X ) of self-homotopy equivalences of elementary A -complexes, exceptthe four-cell Chang-complex C n + , are given by the table, where p is a prime: X E ( X ) Elements S n Z /2 { ± n } M n2 r Aut ( Z /2 r ) ⊕ Z /2 { k · M + ε · i n ηq n + : k ∈ ( Z /2 r ) × , ε ∈ Z /2 } M np r Aut ( Z /p r ) { k · M : k ∈ ( Z /p r ) × } C n + Z /2 ⊕ Z /2 { ± η , 1 η − ˜ ζq n + , − η + ˜ ζq n + } C n + Aut ( Z /2 t + ) ⊕ Z /2 { ± C + y · e ξB n + ( χ ) q M : y ∈ Z /2 t } C n + Aut ( Z /2 r + ) ⊕ Z /2 { ± C + y · i M B n ( χ ) ξ : y ∈ Z /2 r } As for the group E ( C n + ) , we firstly introduce a criterion of detecting self-homotopy equivalencesof C n + , see Lemma 3.1 of [24]. More general, we have Lemma 4.13.
Let X = W ki = C n + i r i , f : X → X . The following statements are equivalent: ( i ) f ∈ E ( X );( ii ) H n ( f ; Z /2 ) ∈ Aut ( H n ( X ; Z /2 )) or H n ( f ; Z /2 ) ∈ Aut ( H n ( X ; Z /2 )) ; ( iii ) H n + ( f ; Z /2 ) ∈ Aut ( H n + ( X ; Z /2 )) or H n + ( f ; Z /2 ) ∈ Aut ( H n + ( X ; Z /2 )) .Proof. It’s clear that H n + j ( X ) ∼ = ⊕ ki = Z /2 r i , j = ; ⊕ ki = Z /2 t i , j = ;
0, j =
0, 1.
By the universal coefficient theorem for cohomology, there is a natural isomorphism: H n + j ( X ; Z /2 ) ∼ = Hom ( H n + j ( X ; Z /2 ) , Z /2 ) ∼ = H n + j ( X ; Z /2 ) . By Lemma 1.1 of [23], there hold equivalences: f ∈ E ( X ) ⇐⇒ H n + j ( f ) ∈ Aut ( H n + j ( X )) , j =
0, 1. ⇐⇒ H n + j ( f ; Z /2 ) ∈ Aut ( H n + j ( X ; Z /2 )) , j =
0, 1, 2. ⇐⇒ H n + j ( f ; Z /2 ) ∈ Aut ( H n + j ( X ; Z /2 )) , j =
0, 1, 2. ( i ) ⇒ ( ii ) is clear. ( ii ) ⇔ ( iii )( ⇒ ( i )) : Consider the commutative diagram: ( ⋆ ) Hom ( H n + ( X ) , Z /2 ) ∼ = / / Hom ( H n + ( f ) , Z /2 ) (cid:15) (cid:15) Ext ( H n + ( X ) , Z /2 ) ∼ = / / Ext ( H n + ( f ) , Z /2 ) (cid:15) (cid:15) H n + ( X ; Z /2 ) H n + ( f ; Z /2 ) (cid:15) (cid:15) ∼ = (cid:15) (cid:15) H n ( X ; Z /2 ) Sq o o ∼ = o o H n ( f ; Z /2 ) (cid:15) (cid:15) ∼ = (cid:15) (cid:15) Hom ( H n + ( X ) , Z /2 ) ∼ = / / Ext ( H n + ( X ) , Z /2 ) ∼ = / / H n + ( X ; Z /2 ) H n ( X ; Z /2 ) Sq o o ∼ = o o where the second isomorphisms in rows follows from the universal coefficient theorem for cohomology,the third isomorphisms in rows hold because the Steenrod square Sq : H n ( X ; Z /2 ) → H n + ( X ; Z /2 ) is a natural isomrphism, by Lemma 2.6. It follows that H n ( f ; Z /2 ) ∈ Aut ( H n ( X ; Z /2 ) ⇐⇒ H n + ( f ; Z /2 ) ∈ Aut ( H n + ( X ; Z /2 )) ⇐⇒ Hom ( H n + ( f ) , Z /2 ) ∈ Hom ( H n + ( X ) , Z /2 ) . OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 29 By the universal coefficient theorem for cohomology again, there is a commutative diagram withshort exact rows: ( ∗ ) H n ( X ; Z /2 ) ∼ = Ext ( H n ( X ) , Z /2 ) / / / / H n ( f ; Z /2 )= (cid:15) (cid:15) Ext ( H n ( f ) , Z /2 ) (cid:15) (cid:15) H n + ( X ; Z /2 ) / / / / H n + ( f ; Z /2 ) (cid:15) (cid:15) Hom ( H n + ( X ) , Z /2 ) Hom ( H n + ( f ) , Z /2 ) (cid:15) (cid:15) H n ( X ; Z /2 ) ∼ = Ext ( H n ( X ) , Z /2 ) / / / / H n + ( X ; Z /2 ) / / / / Hom ( H n + ( X ) , Z /2 ) Then by the snake lemma, the group structures of H n ( X ; Z /2 ) and Hom ( H n + ( X ) , Z /2 ) , the followingequivalences hold: H n ( f ; Z /2 ) ∈ Aut ( H n ( X ; Z /2 ) ⇐⇒ Hom ( H n + ( f ) , Z /2 ) ∈ Hom ( H n + ( X ) , Z /2 ) ⇐⇒ H n + ( f ; Z /2 ) ∈ Aut ( H n + ( X ; Z /2 )) . Thus ( ii ) ⇔ ( iii ) holds and we complete the proof. (cid:3) Corollary 4.14. | E ( W k C n + ) | = | Aut ( ⊕ k Z /2 max ( r,t )+ ) | · k [ · min ( r,t )+ ] , where the group order | Aut ( ⊕ k Z /2 max ( r,t )+ ) | can be computed by Theorem 4.1 of [14].Proof. By the generator set of [ C n + , C n + ] in Table 2, we may set f = [ f ij ] k × k ∈ End ( W k C n + ) ,where f ij = ( x ij , y ij , z ij ) = x ij · C + y ij · i n + q n + + z ij · ω tr for some x ij ∈ Z /2 max ( r,t )+ , y ij ∈ Z /2 min ( r,t )+ , z ij ∈ Z /2 min ( r,t ) , i, j =
1, 2, · · · , k.
By Lemma 4.13, f ∈ E ( W i C i ) if and only if H n ( f ; Z /2 ) = π n ( f ; Z /2 ) = π n ( f ) ⊗ Z /2 ∈ Aut ( ⊕ k Z /2 ) . By Proposition 1.2, we compute that it is equivalent to the identity ¯ f ij ( i n ) ≡ ¯ x ij · i n with ¯ x ij ≡ x ij ( mod ) . The following equivalences hold: f ∈ E ( _ k C n + ) ⇐⇒ [ π n ( f ij ) ⊗ Z /2 ] k × k ∈ Aut ( ⊕ k Z /2 ) ⇐⇒ [ ¯ x ij ] k × k ∈ Aut ( ⊕ k Z /2 ) ⇐⇒ det ([ ¯ x ij ] k × k ) ≡ ( mod ) ⇐⇒ det ([ x ij ] k × k ) ≡ ( mod ) ⇐⇒ [ x ij ] k × k ∈ Aut ( ⊕ k Z /2 max ( r,t )+ ) . Then we get the equality of group order | E ( W k C n + ) by counting admissible x ij , y ij , z ij . (cid:3) We now use the generators and relations to improve Baues’s result about E ( C n + ) by generalizingthe group extension of E ( C n + ) in the case t = r [6, page 53-54] to the case t = r . Proposition 4.15.
Let m = min ( r, t ) , j = max ( r, t ) , ω tr = (cid:12) i M B n ( χ ) ¯ ξq C , t ≥ ri C e ξB n + ( χ ) q M , t < r . E ( C n + ) = (cid:8) x · C + y · i n + q n + + z · ω tr : x ∈ ( Z /2 j + ) × , y ∈ Z /2 m + , z ∈ Z /2 m . (cid:9) ∼ = Z /2 min ( r,t )+ ⋊ (cid:0) Aut ( Z /2 r + ) ⊕ Aut ( Z /2 t + ) (cid:1) . Let ρ a : Z /2 u → Z /2 u be the homomorphism satisfying ρ a ( ) = a . The action of Aut ( Z /2 r + ) ⊕ Aut ( Z /2 t + ) on Z /2 min ( r,t )+ is given by the splitting section s : Aut ( Z /2 r + ) ⊕ Aut ( Z /2 t + ) → E ( C n + ) s ( ρ x , ρ x + ) = ( x, 0, z ) = x · C + z · ω tr ( x odd ) . Proof.
Let C = C n + . By the generator set of [ C, C ] , we may put f = x · C + y · i n + q n + + z · ω tr for some x ∈ Z /2 j + , y ∈ Z /2 m + , z ∈ Z /2 m . By Corollary 4.14, | E ( C n + ) | = min ( t,r )+ t + r + = min ( t,r )+ · | Aut ( Z /2 r + ) | · | Aut ( Z /2 r + ) | . We show the group structure of E ( C ) in the case t ≥ r and omit the similar proof in the case t < r . Let t ≥ r below. Write f = ( x, y, z ) = x · C + y · i n + q n + + z · i M B n ( χ ) ξq C , where x ∈ Z /2 t + , y ∈ Z /2 r + , z ∈ Z /2 r . Then by Proposition 1.2 we get the composition law in [ C, C ] : ( ♯ ) ( x, y, z ) ◦ ( x ′ , y ′ , z ′ ) = ( xx ′ , xy ′ + x ′ y + ′ , xz ′ + x ′ z + ′ ) . Consider the map π : E ( C ) − → Aut ( π n + ( C )) ⊕ Aut ( π n + ( C )) ,π ( f ) = [ π n + ( f ) , π n + ( f )] t is the transpose. Since f ∗ ( q n + ) = k · q n + ( k ∈ Z ) , we have π n + ( f ◦ f ′ ) = π n + ( f ) ◦ π n + ( f ′ ) = π n + ( f ′ ) · π n + ( f ) . Hence π is a homomorphism. Using the coordinate notation f = ( x, y, z ) , we have π ( x, y, z ) = ( ρ x , ρ x + ) . Then we compute that Ker ( π ) = N = Z /2 min ( r,t )+ h (
1, 1, 0 ) i . Note that Aut ( Z /2 u ) ∼ = Z /2 h ρ − i ⊕ Z /2 l − h ρ i ( u ≥ ) , the homomorphism π is surjective.Define s : Aut ( π n + ( C )) ⊕ Aut ( π n + ( C )) − → E ( C ) by setting s ( ρ x , ρ x + ) = ( x, 0, z ) . Then s isa homomorphsm: s (cid:0) ( ρ x , ρ x + )( ρ x ′ , ρ x ′ + ′ ) (cid:1) = s (cid:0) ρ xx ′ , ρ xx ′ + ( xz ′ + x ′ z + ′ ) (cid:1) = ( xx ′ , 0, xz ′ + x ′ z + ′ )= ( x, 0, z ) ◦ ( x ′ , 0, z ′ )= s ( ρ x , ρ x + z ) ◦ s ( ρ x ′ , ρ x ′ + ′ ) . Obviously πs = and hence s is a splitting section of π . Therefore we get a splitting extension: → N → E ( C ) π −− → Aut ( π n + ( C )) ⊕ Aut ( π n + ( C )) → ( π n + ( C )) ⊕ Aut ( π n + ( C )) acts on N by ( x, x + ) ∗ (
1, 1, 0 ) = s ( x, x + ) ◦ (
1, 1, 0 ) ◦ s ( x ′ , x ′ + ′ ) = (
1, 1 + ′ , 0 ) , where ( x ′ , x ′ + ′ ) = ( x, 2z ) − . (cid:3) Remark . (1) The self-homotpy equivalences group E ( X ) of an elementary A -complex X canbe expressed by the following group extension − → Ker ( π ) − → E ( X ) π −− → Aut ( π n + ( X )) ⊕ Aut ( π n + ( X )) − → where π = [ π n + , π n + ] t is the transpose of the homomorphism (cid:18) π n + π n + (cid:19) .(2) If X is a decomposable A -complex, the map π in ( ) is generally not surjective. For example,taking X = S n + ∨ S n + , then E ( X ) ∼ = Gl ( Z ) is the general linear group of dimension ,while Aut ( π n + ( X )) ⊕ Aut ( π n + ( X )) ∼ = Gl ( Z ) ⊕ Gl ( Z ) .Let X = X ∨ X be a wedge of stable spaces, i j : X j → X and q j : X → X j ( j =
1, 2 ) be thecanonical inclusions and projections, respectively. Then a self-map f of X is of the form, in which f ij = p i fi j ∈ [ X j , X i ] : f = (cid:18) f f f f (cid:19) OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 31 Pavesic systematically studied the group of stable self-homotopy equivalences [19, 20, 21]. He definedthat E ( X ∨ X ) is reducible if f ∈ E ( X ∨ X ) implies that f ii ∈ E ( X i ) for i =
1, 2 . The followingtheorem is a restatement of Theorem 2.1 and Proposition 2.5 of [19] and Corollary 4.17 of [21].
Theorem 4.17.
Let X , X be stable CW-complexes satisfying the following two assumptions: ( i ) X is indecomposable and H ∗ ( X ) is a finite (graded) group. ( ii ) X is not a wedge summand of X .Then E ( X ∨ X ) is reducible and admits two decompositions: ( ) the symmetric decomposition: E ( X ∨ X ) ∼ = (cid:0) E ( X ) ⋉ [ X , X ] (cid:1) · (cid:0) E ( X ) ⋉ [ X , X ] (cid:1) , where the semi-products are taken with respect to the action given by the composition from the left. ( ) the LDU-decomposition: E ( X ∨ X ) = (cid:18) [ X , X ] (cid:19) ◦ (cid:18) E ( X ) E ( X ) (cid:19) ◦ (cid:18) [ X , X ] (cid:19) ∼ = [ X , X ] · (cid:0) E ( X ) ⊕ E ( X ) (cid:1) · [ X , X ] Proposition 4.18.
The above theorem holds for X i = C n + i r i , i =
1, 2 and n ≥ .Proof. Let X = X ∨ X . Then X is -local( X ( ) = X ). By Theorem 4.11 of [21], R = [ X, X ] is asemi-perfect ring, which by Corollary 3.17 of [20] implies that R × = E ( X ) admits symmetric andLDU-decompositions as in Theorem 4.17 with respect to the complete idempotent set (cid:12) e = (cid:18) X
00 0 (cid:19) , ¯ e = (cid:18) X (cid:19) (cid:13) . (cid:3) Some natural subgroups.
We need the notations below. Let
X, Y be based CW-complexes, k ≥ . E ∗ ( X ) = { f ∈ E ( X ) : f ∗ = id : H i ( X ) → H i ( X ) for all i } , E k ♯ ( X ) = { f ∈ E ( X ) : f ♯ = id : π i ( X ) → π ( X ) for all i ≤ k } ;[ X, Y ] k ♯ = { f ∈ [ X, Y ] : f ♯ = : π i ( X ) → π i ( Y ) for i ≤ k } , [ X, Y ] k ∗ = { f ∈ [ X, Y ] : f ∗ = : H i ( X ) → H i ( Y ) for i ≤ k } , [ X, Y ] ∗ = [ X, Y ] ∞ ∗ . Note that if X = Σ X ′ , k ≥ k ′ , there are inclusions of subgroups: E k ♯ ( X ) ≤ E k ′ ♯ ( X ); [ X, Y ] k ♯ ≤ [ X, Y ] k ′ ♯ , [ X, Y ] k ∗ ≤ [ X, Y ] k ′ ∗ . Lemma 4.19. If X is an A -complex, n ≥ , then E k ♯ ( X ) is abelian for each k ≥ dimX .Proof. By Cof1 s for elementary Chang-complexes and the fact that C f ∨ g ≃ C f ∨ C g , where C f denotethe homotopy cofiber of f , we see that every A -complex X is a cofiber of a map between wedge sumsof spheres. Then the lemma follows from Corollary 3.5 of [3]. (cid:3) Recall that an ideal I of a unitary ring R is called quasi-invertible if + I ⊆ R × , where R × denotesthe group of units of R . For example, nilpotent ideals are quasi-invertible. Theorem 4.20 (Theorem 4.13 of [21]) . If I is a quasi-invertible ideal of a unitary ring R , then thegroup + I admits a LDU-decomposition with respect to any a complete set of idempotents; in particular, E ∗ ( X ) admits a LDU-decomposition. Lemma 4.21 (Proposition 2.4 of [4]) . Let X be a finite CW-complex, then [ X, X ] dimX ♯ ⊆ [ X, X ] isnilpotent and hence quasi-invertible. Proposition 4.22.
Let n ≥ , X, Y be A -complexes. For any l ≥ , there is a LDU-decomposition: E n + l ♯ ( X ∨ Y ) = [ X, Y ] n + l ♯ ! ⊕ E n + l ♯ ( X ) E n + l ♯ ( Y ) ! ⊕ (cid:18) [ Y, X ] n + l ♯ (cid:19) ∼ = [ X, Y ] n + l ♯ ⊕ (cid:0) E n + l ♯ ( X ) ⊕ E n + ♯ ( Y ) (cid:1) ⊕ [ Y, X ] n + l ♯ . Proof.
By Lemma 4.19, Theorem 4.20 and Lemma 4.21, it suffices to show that E n + l ♯ ( X ∨ Y ) = + [ X ∨ Y, X ∨ Y ] n + l ♯ . The ‘‘ ⊆ " part is clear; for the part ‘‘ ⊇ " , By Theorem 2 of [10], for a finite CW-complex Z , f ∈ E ( Z ) ⇐⇒ f ♯ : π i ( Z ) ∼ = − → π i ( Z ) , i ≤ dim ( Z ) . Hence if l ≥ , the map f = + g is a self-homotopy equivalence for any g ∈ [ X ∨ Y, X ∨ Y ] n + l ♯ ; i.e., f ∈ E n + l ♯ ( X ∨ Y ) . (cid:3) Corollary 4.23.
Let X , · · · , X m be A -complexes, n ≥ . For any l ≥ , we have E n + l ♯ ( X ∨ · · · X m ) ∼ = m M t = E n + l ♯ ( X t ) ⊕ m,m M i = j = [ X i , X j ] n + l ♯ . Example 4.24.
Let X be an elementary Chang-complex, n ≥ . The subgroups E n + ♯ ( X ) are computedas follows. ( ) E n + ♯ ( C n + ) = E n + ♯ ( C n + ) = { } . ( ) E n + ♯ ( C n + ) ∼ = Z /2 . ( ) E n + ♯ ( C n + ) ∼ = Z /2 min ( r,t )+ ⋊ Z /2 , Z /2 h a i acts on Z /2 min ( r,t )+ h b i by a ∗ b = (cid:12) ( + r ) b, t ≥ r ; b, t < r. ; E n + ♯ ( C n + ) ∼ = Z /2 min ( r,t ) ⊕ Z /2 . Proof. If n ≥
4, i ≤ , π n + i ( X ) ∼ = π sn + i ( X ) holds for any an A -complex X . The proof of ( ) is similarbut easier than that of ( ) , so we omit it here. ( ) By Lemma 4.12, we may put f = x · C + y · i M B n ( χ ) ¯ ξ ∈ E ( C n + ) , x = ±
1, y ∈ Z /2 r . Then f ∈ E n + ♯ ( C n + ) is equivalent to x + ≡ ( mod ) r , x =
1, x + ≡ ( mod ) . Hence x =
1, y = r − ε ∈ Z /2 , E n + ♯ ( C n + ) ∼ = Z /2 . ( ) Let C = C n + , ( x, y, z ) = x · C + y · i n + q n + + z · ω tr for some x ∈ Z /2 max ( t,r )+ , y ∈ Z /2 min ( t,r ) , z ∈ Z /2 min ( t,r ) . By Proposition 1.2, we compute that ( i ) If t ≥ r , then ( x, y, z )( i n ) = ( x + ) · i n , ( x, y, z )( i n + ) = x · i n + , ( x, y, z )( i M ˜ η r ) = ( x + ) · i M ˜ η r + y · i n + η. OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 33 Hence ( x, y, z ) ∈ E n + ♯ ( C ) ⇐⇒ (cid:12) x + ≡ ( mod r ) x ≡ ( mod t + ) ⇐⇒ x = ∈ Z /2 r + ,z = r − ε,ε ∈ {
0, 1 } . ( ii ) If t < r , then ( x, y, z )( i n ) = x · i n , ( x, y, z )( i n + ) = ( x + ) · i n + , ( x, y, z )( i M ˜ η r ) = x · i M ˜ η r + y · i n + η. Hence ( x, y, z ) ∈ E n + ♯ ( C ) ⇐⇒ (cid:12) x ≡ ( mod r ) x + ≡ ( mod t + ) ⇐⇒ x = + r ε ′ ,y ∈ Z /2 t + ,z = ′ ∈ {
0, 1 } . Consider the map φ : E n + ♯ ( C ) − → Aut ( π n + ( C )) , φ ( f ) = f ∗ . Since
Aut ( π n + ( C )) ∼ = Aut ( Z /2 r + h q n + i ) , φ ( f ) is a multiple of f , which implies that φ is a homo-morphism of groups. By the above discussion, we haveKer ( φ ) = Z /2 min ( r,t )+ h (
1, 1, 0 ) i , φ ( E n + ♯ )( C ) ∼ = Z /2. One checks that a splitting section s : Z /2 − → E n + ♯ ( C ) of π can be given by s ( ) = (cid:12) (
1, 0, 2 r − ) , t ≥ r ( + r , 0, 0 ) , t < r We conclude that E n + ♯ ( C n + ) = (cid:14) (cid:8) C + y · i n + q n + + r − ε · i M B n ( χ ) ξq C (cid:9) , t ≥ r ; (cid:8) ( + r ε ′ ) · C + y · i n + q n + (cid:9) , t < r. ∼ = Z /2 min ( r,t )+ ⋊ Z /2. By ( i ) and ( ii ) , ( x, y, z ) ∈ E n + ♯ ( C ) induces the identity on π n + ( C ) if and only if y ∈ h i ≤ Z /2 min ( t,r )+ . Thus E n + ♯ ( C ) ∼ = Z /2 min ( r,t ) ⊕ Z /2 , whose generator set can be chosen to be { (
1, 2, 0 ) , (
1, 0, 2 r − ) } if t ≥ r ; { (
1, 2, 0 ) , ( + r , 0, 0 ) } if t < r . (cid:3) Lemma 4.25.
Let n ≥ , X, Y be A -complexes with H n + ( X ) = H n + ( Y ) = . ( ) For any l ≥ , [ X, Y ] n + l ♯ ( X ) ≤ [ X, Y ] n + ∗ = [ X, Y ] ∗ . ( ) E n + ♯ ( X ) E E ∗ ( X ) ≤ E n ♯ ( X ) . Proof.
Suppose that f ∈ [ X, Y ] and consider the following map of Γ -sequences: / / Γ ( H n ( X )) / / Γ ( H n ( f )) (cid:15) (cid:15) π n + ( X ) h n + / / π n + ( f ) (cid:15) (cid:15) H n + ( X ) H n + ( f ) (cid:15) (cid:15) / / / / Γ ( H n ( Y )) / / π n + ( Y ) h n + / / H n + ( X ) / / where Γ ( H n ( X )) = H n ( X ) ⊗ Z /2 . ( ) For any an A -complex X with H n + ( X ) = , by Hurewicz Theorem, the Hurewicz homomor-phism h n : π n ( X ) → π n ( X ) is an isomorphism and h n + : π n + ( X ) → H n + ( X ) is a surjection. By thecommutativity of the above diagram, π n + ( f ) = implies H n + ( f ) = , hence [ X, Y ] n + l ♯ ≤ [ X, Y ] n + l ∗ for l =
0, 1 . ( ) The second ‘‘ ≤ ′′ is clear. Suppose that f ∈ E n + ♯ ( X ) . By Hurewicz theorem and the com-mutative diagram in ( ) with Y = X , π n + i ( f ) = id implies that H n + i ( f ) = id , i =
0, 1 . Hence f ∈ E n + ∗ ( X ) = E ( X ) . (cid:3) Proposition 4.26.
Let n ≥ , X, Y be A -complexes with H n + ( X ) = H n + ( Y ) = . Then E n + ♯ ( X ∨ Y ) admits a LDU-decomposition: E n + ♯ ( X ∨ Y ) = [ X, Y ] n + ♯ ! · E n + ♯ ( X ) E n + ♯ ( Y ) ! · (cid:18) [ Y, X ] n + ♯ (cid:19) ∼ = [ X, Y ] n + ♯ · (cid:0) E n + ♯ ( X ) ⊕ E n + ♯ ( Y ) (cid:1) · [ Y, X ] n + ♯ . Proof.
By Lemma 4.25 ( ) , E n + ♯ ( X ∨ Y ) = + [ X ∨ Y, X ∨ Y ] n + ♯ ≤ E ∗ ( X ∨ Y ) ; by Lemma 4.25 ( ) andWhitehead theorem, [ X ∨ Y, X ∨ Y ] n + ♯ is a quasi-invertible ideal of End ( X ∨ Y ) . Then the theoremfollows from Theorem 4.20. (cid:3) We end up this section by computing some natural subgroups of the self-homotopy groups of W = C n + i r i and determining relations between E ∗ ( W = C n + i r i ) and E n + l ♯ ( W = C n + i r i ) for l =
1, 2 . Lemma 4.27.
Let n ≥
4, C = C n + , C ′ = C n + ′ r ′ . ( ) [ C, C ′ ] n + ♯ ∼ = Z /2 min ( r,t ′ ) ⊕ Z /2 . ( ) [ C, C ′ ] ∗ ∼ = [ C, C ′ ] n + ♯ ⊕ Z /2 .Proof. ( ) If n ≥ , π n + j ( C ) ∼ = π sn + j ( C ) , j =
0, 1, 2 . The computations can be separated into threecases. ( i ) t = t ′ ∧ r = r ′ . f = ( x, y, z ) = x · C + y · i n + q n + + z · ω tr ∈ [ C, C ] n + ♯ if and only if y ∈ h i ≤ Z /2 min ( t,r )+ ∧ (cid:12) x =
0, z = r − ε, t ≥ rx = r ε ′ , z =
0, t < r
Hence [ C, C ] n + ♯ ∼ = Z /2 min ( t,r ) ⊕ Z /2 . ( ii ) t ′ ≥ t ∧ r ′ < r or t ′ > t ∧ r ′ ≤ r . Recall that in this case we have [ C, C ′ ] ∼ = Z /2 max ( t,r ′ )+ h i C ′ e κ i ⊕ Z /2 min ( t ′ ,r )+ h i n + q n + i ⊕ Z /2 min ( t,r ′ ) h ω tr ′ i . If t ′ ≥ t ∧ r ′ < r , there hold relations: ( R ) q ′ M ◦ i C ′ e κ = B n + ( χ ) ◦ q M , i C ′ e κ ◦ i n + = t ′ − t · i n + .i C ′ e κ ◦ i M = i ′ M ◦ B n ( χ ) , i C ′ e κ ◦ i n = i n .i ′ M B n ( χ ) ξq C ◦ i n = · i n .i M B n ( χ ) ξq C ′ ◦ i n = (cid:12) · i n , r ′ ≥ r − r − r ′ · i n r ′ < r − C e ξB n + ( χ ) q ′ M ◦ i n + = (cid:12) i n + , t ′ > t2 · i n + t ′ = t . If t ′ ≥ t ≥ r ′ < r , put f = ( x, y, z ) = x · i C ′ e κ + y · i n + q n + + z · i ′ M B n ( χ ) e ξq C ∈ [ C, C ′ ] . Then by computations, [ C, C ′ ] n + ♯ = { (
0, y, 2 r ′ − ε ′ ) : y ∈ h i ∈ Z /2 min ( r,t ′ )+ }∼ = Z /2 min ( r,t ′ ) ⊕ Z /2. Other cases can be similarly proved.
OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 35 ( iii ) t ′ < t ∨ r ′ > r . Put f = ( x, y, z ) = x · i n + q n + + y · i ′ M B n ( χ ) ξq C + z · i C ′ e ξB n + ( χ ) q M for some x ∈ Z /2 min ( r,t ′ )+ , y ∈ Z /2 min ( r + ′ ) , z ∈ Z /2 min ( t,t ′ + ) . We only give the proof inthe case t > t ′ ≥ r < r ′ here. In this case there hold relations: ( R ) i C e κ ◦ i ′ M = i M B n ( χ ) , i C e κ ◦ i n = i n .q M ◦ i C e κ = B n + ( χ ) q ′ M , i C e κ ◦ i n + = t − t ′ · i n + i M B n ( χ ) ξq C ′ ◦ i n = · i n .i ′ M B n ( χ ) ξq C ◦ i n = r ′ − r · i n .i C e ξB n + ( χ ) q ′ M ◦ i n + = t + − t ′ · i n + .i C ′ e ξB n + ( χ ) q M ◦ i n + = i n + . It is easy to check that f ∈ [ C, C ′ ] n + ♯ ⇐⇒ x ∈ h i ∈ Z /2 r + , y = r ε, z = Hence [ C, C ′ ] n + ♯ ∼ = Z /2 r ⊕ Z /2 . ( ) By Lemma 4.25, there is an exact sequence: → [ C, C ′ ] n + ♯ → [ C, C ′ ] ∗ π n + −−−− → Hom ( Z /2 t + , Z /2 t ′ + ) . By similar computations as in ( ) , one can show that Im ( π n + ) ∼ = Z /2 and there is a splitting section Z /2 → [ C, C ] ∗ . Thus [ C, C ′ ] ∗ ∼ = [ C, C ′ ] n + ♯ ⊕ Z /2 ; the detailed computations are omitted here. (cid:3) Theorem 4.28. n ≥ E n + ♯ ( W ki = C n + i r i ) ∼ = L k,ki,j = Z /2 min ( r i ,t j ) ⊕ ( Z /2 ) s .Proof. By Lemma 4.19, Example 4.24 and Lemma 4.27. (cid:3)
Let C i = C n + i r i , i =
1, 2 . For the remainder of this section we study the relations among E ∗ ( C ∨ C ) , E n + l ♯ ( C ∨ C ) , l =
1, 2 . Theorem 4.29.
There is a splitting group extension: − → M i,j = Z /2 min ( r i ,t j )+ − → E n + ♯ ( C ∨ C ) π n + −−−− → ( Z /2 ) − → Proof.
We only give the proof in the case t ≥ t ≥ r ≤ r here; other cases can be similarlycomputed.Let y ij ∈ Z /2 min ( r i ,t j )+ , ε ij ∈ {
0, 1 } . If t ≥ t ≥ r ≤ r , by Proposition 4.26, we may put f = [ f ij ] × , in which f ii ∈ E n + ♯ ( C i ) , f ij ∈ [ C j , C i ] n + ♯ ( i = j ) . By the proof of Lemma 4.27, it is easyto compute that elements of E n + ♯ ( W si = C i ) are of the form: f = " (
1, y , 2 r − ε ) (
0, y , 2 r − ε )(
0, y , 2 r ε ) (
1, y , 2 r − ε ) . Consider the map induced on cohomotopy groups: π n + : E n + ♯ ( C ∨ C ) − → Aut (cid:0) π n + ( C ∨ C ) (cid:1) ,π n + ( f ) = f ♯ ( n + ) . By computations we haveIm ( π n + ) = (cid:14) " + r ε r ε r ε + r ε (cid:15) ; Ker ( π n + ) = (cid:14) " (
1, y , 0 ) (
0, y , 0 )(
0, y , 0 ) (
1, y , 0 ) (cid:15) The composition law in Im ( π n + ) is given by [ δ ij + r j ε ij ] · [ δ ij + r j ε ′ ij ] = [ δ ij + r j ( ε ij + ε ′ ij )] , hence Im ( π n + ) is abelian and isomorphic to ( Z /2 ) , which also implies that the map π n + is ahomomorphism of groups: π n + ( f ◦ f ′ ) = ( f ◦ f ′ ) ♯ ( n + ) = f ′ ♯ ( n + ) · f ♯ ( n + ) = f ♯ ( n + ) · f ′ ♯ ( n + ) = π n + ( f ) · π n + ( f ′ ) . The composition law in Ker ( π n + ) is given by [( δ ij , y ij , 0 )] ◦ [( δ ij , y ′ ij , 0 )] = [( δ ij , y ij + y ′ ij , 0 )] . Hence Ker ( π n + ) is abelian and Ker ( π n + ) ∼ = L = Z /2 min ( r i ,t j )+ . We now give a splitting section of π n + , which completes the proof. Denote element of Im ( π n + ) by E = " (
1, 0, 2 r − ε ) (
0, 0, 2 r − ε )(
0, 0, 2 r ε ) (
1, 0, 2 r − ε ) , denote E ′ the new map by replacing ε ij with ε ′ ij . We compute that E ◦ E ′ = " (
1, 0, 2 r − ( ε + ε ′ )) (
0, 0, 2 r − ( ε + ε ′ ))(
0, 0, 2 r ( ε + ε ′ )) (
1, 0, 2 r − ( ε + ε ′ )) Define s : Im ( π n + ) → E n + ♯ ( C ∨ C ) by setting s (cid:0) [ δ ij + r j ε ij ] (cid:1) = E , then one checks that s is ahomomorphism such that π n + ◦ s = id . (cid:3) Theorem 4.30.
There is a group extension: − → E n + ♯ ( C ∨ C ) − → E ∗ ( C ∨ C ) π −− → ( Z /2 ) ⊕ ( Z /2 ) − → where E n + ♯ ( C ∨ C ) ∼ = L = Z /2 min ( r i ,t j ) ⊕ ( Z /2 ) is computed by Theorem 4.28.Proof. We only give the proof in the case t ≥ t ≥ r ≤ r here and omitted other similar cases.Consider the homomorphism π : E ∗ ( C ∨ C ) −− → M i = Aut (cid:0) π n + i ( C ∨ C ) (cid:1) , with π ( f ) = [ π n + ( f ) , π n + ( f )] t . Let G i = π n + i (cid:0) E ∗ ( C ∨ C ) (cid:1) , i =
1, 2 . By Lemma 4.25 ( ii ) , there isa group extension − → E n + ♯ ( C ∨ C ) − → E ∗ ( C ∨ C ) π −− → G ⊕ G − → By Theorem 4.20 we may put f = [ f ij ] × ∈ E ∗ ( C ∨ C ) , with f ij ∈ E ∗ ( C i )( i = j ) , f ij ∈ [ C j , C i ] ∗ ( i = j ) . Then π n + ( f ) = (cid:0) π n + ( f ij ) (cid:1) . Similar to the proof of Lemma 4.27, put f ij = ( x ij , y ij , z ij ) = x ij · i C ′ ˜ κ + y ij · i n + q n + + z ij · i ′ M B n ( χ ) ¯ ξq C , for some x ij ∈ Z /2 t j + , y ij ∈ Z /2 r i + , z ij ∈ Z /2 r i . Recall that π n + ( C i ) ∼ = Z /2 t i + { i n + } , π n + ( C i ) ∼ = Z /2 ⊕ Z /2 { i n + η, i M ˜ η r i } . Let ε ij , ¯ ε ij ∈ {
0, 1 } . i = j . If t i ≥ r i , then x ij ∈ Z /2 t i + , y ii ∈ Z /2 r i + , z ij ∈ Z /2 r i , f ii ∈ E ∗ ( C i ) ⇐⇒ x ii = + t i ε ii , z ii = r i − ˜ ε ii . OMOTOPY CLASSIFICATION OF MAPS BETWEEN A -COMPLEXES 37 Hence by relations ( R ) we get π n + ( f ii ) = + t i , π n + ( f ii ) = (cid:18) ii (cid:19) .i = j . f ij ∈ [ C j , C i ] ∗ ⇐⇒ x ij = t i ε ij , z ij ∈ r i − ¯ ε ij . hence π n + ( f ij ) = t i ε ij , π n + ( f ij ) = (cid:18) ij (cid:19) . Thus we get G = (cid:14) (cid:2) δ ij + t i ε ij (cid:3) × ∈ Aut ( M i = Z /2 t i + ) (cid:15) ; G = (cid:14) (cid:20)(cid:18) δ ij y ij ij (cid:19)(cid:21) × ∈ Aut ( Z /2 ⊕ Z /2 ) ) (cid:15) . The composition law in G is given by [ δ ij + t i ε ij ] · [ δ ij + t i ε ′ ij ] = [ δ ij + t i ( ε ij + ε ′ ij )]; while the composition law in G is given by (cid:20)(cid:18) δ ij y ij ij (cid:19)(cid:21) · (cid:20)(cid:18) δ ij y ′ ij ij (cid:19)(cid:21) = (cid:20)(cid:18) δ ij y ij + y ′ ij ij (cid:19)(cid:21) . Thus G , G are both abelian groups and isomorphic to ( Z /2 ) .Define a map s : G − → E ∗ ( C ∨ C ) by setting s ([ δ ij + t i ε ij ]) = [( δ ij + t i ε ij , 0, 0 )] , then onechecks that s is a homomorphism of groups and satisfies π ◦ s = G . (cid:3) Corollary 4.31. E ∗ ( C ∨ C ) ∼ = E n + ♯ ( C ∨ C ) ⋊ ( Z /2 ) .Proof. By the proof of Theorem 4.30, if t ≥ t ≥ r ≤ r , there is a splitting group extension / / E n + ♯ ( C ∨ C ) / / E ∗ ( C ∨ C ) π n + / / ( Z /2 ) n n / / Other cases can be similarly computed and the details are omitted here. (cid:3)
Acknowledgements
The author would like to thank Professor Jianzhong Pan for his kind guidance during my PhDstudy and for revising some mistakes in this article.
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